(written from a Production point of view)
The design of the Klingon K't'inga-class, premiering in Star Trek: The Motion Picture, was based on the Koro-class battle cruiser for the movie's immediate unrealized predecessor Star Trek: Phase II, which in turn was based on the D7-class, created by Matt Jefferies for Star Trek: The Original Series, which was itself based on the manta ray in both shape and color. (Star Trek: The Original Series Sketchbook) The studio model originally started out as belonging to the latter class. While the K't'inga-class, like its D7-class predecessor, had only made a limited amount of appearances, together they have remained the quintessential Klingon starship design.
The physical model
The studio model started out life as the The Motion Picture D7/Koro-class model, constructed at Magicam in the spring of 1978 under supervision of Jim Dow. The model itself was subsequently upgraded at Astra Image Corporation (ASTRA), the subsidiary of visual effects (VFX) company Robert Abel & Associates (RA&A), when it was deemed unsuitable to meet big screen requirements after initial test shooting in July 1978. The six foot model was still residing at the ASTRA's Seward St, Los Angeles, studio model effects filming site, when the February 1979 VFX debacle occurred. While considerable surface re-detailing had already been done on the model at ASTRA, Douglas Trumbull set to work a newly appointed Animation and Graphics Artist, Leslie Ekker, to do some additional redesign work on the Klingon battle cruiser, when his company Future General Corporation (FGC) took over the effects photography from RA&A early in the subsequent month,
"My initial duties on the job were to design little details on the Klingon engine pods that they were contemplating changing. Andy Probert wanted some changes done and asked me to do some sketches, because of the way the ship had been designed, I believe, by someone at Astra. There was a kind of an observation deck-appearing detail that looked like a rounded bank of windows in the leading edge of the engine nacelles.They looked like windows and it was ridiculous, because there were no personnel areas in those parts of the ship according to Andy's realistic plan. It just looked wrong. and it looked old-fashioned. We executed those changes, among many other small details on the cruiser that took up a lot of time. Andy got in a lot of hot water because of what I would call his integrity, really. You see, he's a very honest designer and he does know a certain amount about realistic spaceship design. If he sees something contradictory to reality then he will try and change it. And it got to be a power struggle after a while, because Andy would indeed see things that needed changing and there just wasn't time or money. The Klingon cruiser ended up costing 80 or 90 thousand dollars. (Return to Tomorrow - The Filming of Star Trek: The Motion Picture, pp. 200-201)
Ekker's work notwithstanding, essentially shown in the movie as featured in the "paint-scraping" fly-over detail close-ups of the Klingon cruisers in the opening sequences of the movie, it was not yet elaborate enough as far as Apogee, Inc., John Dykstra's VFX company brought in by FGC very shortly after they had acquired the commission, was concerned when the model was delivered to its San Fernando Valley effects stage. Further upgrades on the model were therefore undertaken in late March 1979 by Apogee's model shop, when that company was handed over the studio model. Dykstra recalled, "We had to modify it pretty severely though, because apparently it was not designed to be shot in the same kind of circumstances that we ended up shooting it in. I don't know what Bob Abel planned to do with it, but for us, the practical lighting on the model was so dim that we weren't able to get a good exposure off it even by pushing the film a couple stops and using a twenty-second exposure", Dykstra elaborated (Cinefex, issue 2, p. 52), having added further,
"That model, when it came to us, had been set up for some other photographic technique that didn't fit with ours. So we had to go in and completely redo the lighting. That was done by Grant McCune and the people he had working with him, which is basically a very similar staff to the ones we had on Star Wars and other shows. They did an incredible job. They took all the teeny little lights out of the model and put in some lights of significant size – and they did it in a matter of a few days. They really did a nice job of redoing the model, without destroying what was already there. They had to add a lot of detail to it because we got much closer to the model than I think they ever intended to do before. Once the thing with the ship was worked out, Doug Smith, in charge of shooting that sequence, photographed the opening shot." (American Cinematographer, February 1980, p. 174)
While never referenced as such in neither the movie nor the script, in that process the D7/Koro-class became the K'tinga-class, coined as such by the movie's Executive Producer Gene Roddenberry in his novelization of the movie, written at the time. As indicated, the most significant change in the design of the K't'inga model was its more detailed surface, so that it would look more believable on the movie screen. The K't'inga-class cruiser sequence shots for Star Trek: The Motion Picture were shot under the supervision of Dykstra. (Cinefex, No.1, 1980, pp. 4-34) The footage taken for this movie were the only ones taken of the model in this finish.  Later appearances in Star Trek II: The Wrath of Khan and Star Trek: The Next Generation were stock footage taken from this shoot.
According to Ekker, the construction costs of the model amounted in the end to a total of between US$80,000 and $90,000. (Return to Tomorrow - The Filming of Star Trek: The Motion Picture, p. 201)
For Star Trek VI: The Undiscovered Country the original studio model underwent extensive modification in 1991 for it to become Gorkon's flagship Kronos One. Originally, a new design for Gorkon's ship was considered, but budget restrictions, due to the dismal box-office performance of the previous movie outing, Star Trek V: The Final Frontier, made this impracticable, as Industrial Light & Magic (ILM)'s model maker Bill George recalled, "Paramount had already done some design work on this ship. Mark Moore and I did some additional sketches and I actually built a prototype. But it was ultimately decided that there wasn't money for this, so instead they decided to use the Klingon battlecruiser from Star Trek: The Motion Picture." That entailed more work than George had anticipated as the model had suffered from years of storage, but he made use of the opportunity to perform modifications. These, done at ILM's model shop by him, Moore, and John Goodson, involved a new paint scheme and gold colored metal etched ornaments applied all over the model.  George further elaborated:
"Originally, it was dark green in color; but after uncrating the model, we found it had been covered with a white water-based paint. It looked like someone had tried to clean it off, but that hadn't worked. Since we were going to have to repaint it, we thought that even if it wasn't a new model this was still going to be a special ship, so we figured we'd make it look different – distinctive from what we'd seen in previous films. Mark and I sent out to the library for books on military hardware; and out of that research, we came up with the concept that when Klingons return victorious from battle, they add some new piece of equipment or new graphics to their ships. We chose medieval armor as both our color palette and our design springboard and devised this regal and ornate looking paint job, which everyone liked and approved, so we transferred that look onto the model." (Cinefex, no.49, 1992, p.49)
In addition the warp engines were outfitted with internal lighting visible through newly-applied jagged slotted patterns to the hull. George was exceedingly pleased with the end result, having remarked on another occasion, "It was one of the few models we could alter to look new for this show. After we did some research into military costuming, we came up with the concept that when these ships return victorious from battle, the Klingons build some sort of epaulet onto their wings. we added these golden etched brass epaulets, based on some of the helmet designs we'd seen. It looks very regal, and contrasts nicely with the Enterprise when they are flying together. The Enterprise is smooth, monochromatic and cool, while this Klingon ship is adorned, ostentatious and warm" (Cinefantastique, Vol 22 #5 1992, p. 49)
Upon completion of The Undiscovered Country, the model was almost immediately loaned out to the Smithsonian Institution for their Star Trek Smithsonian Exhibit, where it was shortly re-united with its "mother", the original D7-class model.  It went on to be included in the February 1995 opening leg of the Star Trek: The Exhibition tour in Edinburgh, Scotland.
Yet, visual effect supervisors of the ongoing Star Trek productions had more uses for it in mind, resulting in that the model in this finish was used twice more. The first time was for Star Trek: Deep Space Nine's seasons four double episode, "The Way of the Warrior". Returning in late May 1995 with the Negh'Var model, also slated to feature heavily in the episode, from the The Exhibition, Effects Supervisor Gary Hutzel was dismayed when he discovered that both models were severely damaged, "Apparently when the Neg'Var model got to customs they thought there might be drugs hidden in it so they broke it open. When it came back it was broken in pieces. That was a nasty surprise. The Klingon heavy cruiser was also damaged. We had to quickly make Styrofoam mock ups of those ships and shoot around them until we could have the repairs done." Needing a multitude of Klingon ships for that production, Co-Effects Supervisor Glenn Neufeld, exclaimed at the time, "We had a kit of the old-style cruiser and shot the full size cruiser that Industrial Light and Magic [sic.] built for the features." (Cinefantastique, Vol 28 #4/5, 1996, p. 72)
A design patent, No. D263856, was issued by the US Patent and Trademark Office for the K't'inga on 4 April 1982 (there called a "toy spaceship") noted Andrew Probert as the sole "inventor" of the design. Filed on 7 May 1979, the patent, after being issued, was valid for fourteen years.
The original physical studio model was eventually listed as Lot 996 in the 40 Years of Star Trek: The Collection auction, where it was given an estimated sale price of US$3,000 to $5,000; it ultimately sold for US$85,000 ($102,000 with premium) on 7 October 2006. The model was acquired by Microsoft's co-founder Paul Allen for his Science Fiction Museum and Hall of Fame in Seattle, now known as the Museum of Pop Culture (MoPop).  After its purchase, the museum hired Ed Miarecki's company, FX Models, to do restoration work and to build a custom made display cabinet for the model. It went on public display in the Star Trek: Exploring New Worlds exhibition at MoPop from 21 May 2016 to 28 May 2018 , which subsequently became a touring exhibition.   
Filming the physical model
Apogee did not only modify the studio model, but was also responsible for shooting and compositing the opening Klingon-V'Ger sequence of The Motion Picture. Upon completion of the modifications, the model was delivered to Apogee's filming stage facility, at the time located in the San Fernando Valley, and John Dykstra started to ponder how to go about the sequence,
"So there were Klingon warships moving around in space with no cloud to go with them [rem: that visual effect had not been produced yet]. We got a model of the Klingon ship and looked at it to see what we could do with the bloody thing. We figured out some moves and storyboarded them and went over them with Bob Wise to see if he would go for them or not. We had to go over them with Doug [Trumbull], as well, because Doug, at the time, had the intention of generating the cloud in a way that was different from how it was finally generated. We provided moves that were very slow and deliberate, because Bob wanted to preserve a mass look in terms of the ships. Because of the way the cloud was to be generated, you couldn't move the ships very radically, the reason being that you couldn't duplicate that kind of move on the background. So it was hot using a conventional system–blue screen with one model." (American Cinematographer, February 1980, p. 174)
Once Dykstra had figured out the methodology, his crew of effects camera men went to work. Dykstra further recalled,
"We used the one Klingon model for the whole sequence–all bluescreen composite combination, except for the opening shot which was a double-pass matte deal. We used two passes on that to get a high-con matte because we couldn't do the roll-over trick and keep the bluescreen behind the model.
"Doug Smith was pretty much responsible for working that out. The whole idea was to come up with something that was nice and yet unusual. People are always coming to us and saying they don't want an opening shot that looks like Star Wars–they just want something that has the same impact. Well, that gets a little tough after awhile. The Star Wars opener was incredibly dramatic. It showed you the scale and the size of those ships, and it gave you a sense of speed that you don't normally get in a space environment. So it's an effective trick, but it's since been overused. The idea with the Klingons was to give us the same kind of visual impact, but without doing a real close flyby. (Cinefex, issue 2, p. 52)
"The cloud was shot with stars, which made it a composite element when it went into the optical printer to be put together with the other elements. All of the ships were separations. Some of them were run with cover mattes, and some without, depending upon how many passes there were. It was kind of tough getting the stars to hold up in some of them." (American Cinematographer, February 1980, p. 174)
In regard to the opening sequence, Dykstra later elaborated, "This was an interesting shot, because it was sort of the signature piece that I used to convince Mr. Roddenberry that we could do a big opening for the show. Now, the idea was to be able to do this huge camera move, rotating over the top of the ship and ending up behind it. An extremely long opening shot. And it was a very difficult shot to do. I took almost three weeks of original photography to get the single center ship. (...) Of course, there was only one Klingon ship. So, we had to rephotograph and redress the single ship for three different variations, so you felt as though you were looking at three different ships" (Star Trek: The Motion Picture (The Director's Edition) DVD, "audio commentary")
Originally, the finale of the Klingon-V'Ger encounter was scripted in the latest script treatments of the direct precursor of The Motion Picture, the movie-upgrade of Star Trek: Phase II's "In Thy Image", as ending with the destruction of the Klingon vessels by V'Ger. To this end Magicam had breakaway models of the D7-class predecessor of the K't'inga-class studio model constructed, and these were already blown up and filmed at Robert Abel & Associates. By the time the project had become The Motion Picture, the sequence had changed to become the digitization of the vessels for storage in V'ger' databanks, and Dykstra had to come up with new visual effects to visualize the new ending.
In the movie, the Klingon ships, are hit by V'ger's energy bolts, they themselves called "Whiplash Bolts" by Dykstra and created by a Tesla coil, and amid energy discharges evaporate, instead of blowing up. For the part were the ships are hit by lightning, also generated by a Tesla coil, additional model sculptures were constructed as Dykstra explained, "The real electrical discharge shows up in the Klingon sequence where the cloud fires it energy weapon and that great ball of blue energy comes out and envelopes the ship with electrical discharge all over its surface. It is real lightning done on a form that exactly matches the contours of the ship. Bruno George and David Sosalla sculptured a lot of those forms in the same shape as the ship. They matched them up in the format and then photographed the electric discharge actually crawling up over the surface of the forms." (American Cinematographer, February 1980, p. 195)
The digitization and disappearances of the ships was an even more time consuming and elaborate process. Dykstra further explained,
"Then we went to a laser scan technique to create the disappearance, or digitization, of the Klingon ships. We used a smaller laser for that. It was five watts–which is still a pretty potent laser, but the problem is that when you start scanning it, you're essentially diluting the energy that it's putting out over a larger area and you end up with a limited amount of light. So the exposures had to be quite long. At first we'd scan just a small part of the model; and then we'd increase the scanned area on each subsequent frame so that the digitization effect would appear to be sweeping over the ship. The lines start out far apart and then get closer and closer together as they move across the surface. Then at the end there's something like an arc–a brilliant, flaring light which is supposed to indicate that the ship's molecular structure has been reduced to raw information and consumed by the cloud. It's been destroyed, but in a more elegant way than just blown up. It's violent and destructive, but the kicker–which you don't find out until later–is that the digitization is really a method of information storage. "To get the effect we wanted, we had two, and sometimes more, scanning passes to build up the image. We scanned the model itself; then we cover the model with foil and did another pass. That way we had a hard line for the initial scan and then a sparkling line following it. If the scan line wasn't coming out bight enough on one pass, we'd have to do a second. The we superimposed the scanned footage over the footage of the model itself, which was also in movement at the time. So it was a pretty complex process, and the scanning device that made it all possible was put together on very short notice by Al Miller and his electronics crew. After that was done, Harry Moreau and John Millerburg and the people up in the rotoscope department made frame-by-frame roto mattes so the ship would fade out and disappear." (Cinefex, issue 2, p. 56)
For all the complexity and time expenditure, Dykstra and his crew had to do it all over again with the Epsilon IX station enlarged panel model for its demise later on in the movie.
In the "Special Features" segment on disc seven of the VOY Season 3 DVD entitled Flashback to "Flashback", there is a 0:02:45 segment with Dan Curry discussing the filming at Image G (where the footage of this and the other physical filming models was normally shot for the television productions) of the encounter between the USS Excelsior and Kang's battle cruiser in the Azure Nebula for the Voyager episode "Flashback".
Other physical models
Though the K't'ingas shown in The Next Generation were all stock-footage taken from The Motion Picture, there was one exception with the appearance of a K't'inga-class cruiser at the Surplus Depot Z-15 in "Unification I", where a shot of the ship was used not covered by the footage. For the shot Gregory Jein, assisted by Bruce MacRae and Scott Schneider, built a separate model, using a set of the 1977 three feet molds of the D7 Jein owned.  This model was on tour in 1996 and 1997, appearing amongst others in the 1996 LA Star Trek Convention and in the Boss Film Studios' model shop expo '97 in Los Angeles and seemed, a bit surprisingly perhaps considering the effort that went into the later "The Way of the Warrior" episode, not to have been used since in subsequent Star Trek productions. (Sci-Fi & Fantasy Models, issue 23, October 1997, p. 47)
In "Way of the Warrior" several other physicals models of the K'tinga were used for the massive battle scene, supervised by Dan Curry, Gary Hutzel, and Glenn Neufeld. These however were modified commercially available Playmates Toys ("We had to take out the sound effects", according to Judy Elkins), Hallmark Christmas ornaments and AMT/Ertl Star Trek model kits ("which blew up real good"). (Cinefantastique, Volume 28, No.4/5, p. 72; Star Trek: Deep Space Nine Companion, p. 263) Being the first mass battle scene ever shown on Star Trek, it was also the last one entirely done with physical models (save for a few shots with a CGI Defiant, the only one available at the time). After one of these models had made an encore in the later fourth season episode "Rules of Engagement", Visual Effects supervisor David Stipes had the procedure repeated with model kits for Deep Space Nine's fifth season episode "Call to Arms", the next-to-last time physical models of the K't'inga were used, having stated the time, " I'm not sure what K't'inga is. There are a lot of old style Klingon D-7 [sic.] Cruisers and Vor'chas. (D-7's are model kits and Vor'chs's [sic.] are the sound making toys.)". 
Repeating what he had previously done with the IKS Drovana model, the heavily battle-damaged IKS B'Moth as seen in the earlier episode of that season, "Soldiers of the Empire", was yet another separately-built model by Greg Jein, again having used his set of the 1977 three feet molds of the D7. This model was on tour in 1997 shortly after filming and could be seen at the Star Trek Convention in Pasadena, 1997. 
One of the AMT model kits that David Stipes had used, made an additional appearance for the opening retreating Second Fleet scene in the Deep Space Nine season six opening episode, "A Time to Stand". It joined the multitude of kit-bashed models of Federation ships, that were constructed by several members of the production staff, in order to beef out the scene. One of them, Adam Buckner, has explained that the model was built in haste at the time, and had therefore neither received a paint-job nor an internal lighting rig. Paint and lighting were for its re-use added in post-production.  The time restraints also explained why the large original studio model was not used as rigging that model (and therefore its resultant shuttling of VFX supervisor to and fro between studio and Image G), was too time consuming. Its use represented the last time a physical model of the K't'inga-class was used for representation of the vessel in the franchise, having been executed as CGI for its subsequent appearance in "The Changing Face of Evil".
A CGI model of a K't'inga-class vessel, in fact one of the two very first CGI ships ever for Star Trek, had already been built as early as 1981, but has not been seen by the general public. It was constructed as a probation piece at the Graphics Group to convince mother company ILM and Paramount Pictures to go ahead with what would eventually become the "Genesis Demo" in The Wrath of Khan. Project leader Alvy Ray Smith recalled, "It should be mentioned that we had already done one "mini-production" (in August '81) for ILM to show off our abilities. We had rendered a model of the Starship Enterprise chasing a Klingon ship in 3-dimensions, with appropriate lighting, coloring, etc. This was done using Loren's first working version of his 3-D rendering program "Reyes" (Renders Everything You Ever Saw). This mini-production had also featured several 2-D effects programmed by Pat Cole and Rob Cook. We believe that this demonstration was what prompted Paramount and ILM to ask us to do a piece of the final film." (American Cinematographer, October 1982, p. 1038)
A low resolution CGI model of the K't'inga-class was build years later at Foundation Imaging by Jose Perez for later appearances in Deep Space Nine and the Star Trek: Voyager episode "Prophecy" (although it was supposed to be a D7-class vessel), making its debut in "The Changing Face of Evil". The CGI model was later enhanced by Adam Lebowitz and Robert Bonchune for representation in their book Star Trek: Starship Spotter. The CGI model made subsequent in-print appearances in the licensed Star Trek: Ships of the Line calendars, their book derivatives, and, most conspicuously and further refined, in the British Star Trek: The Official Starships Collection partwork publication.
Apart form its "misuse" in the Voyager episode, the model was also "misused" as Vorok's battle cruiser in the later Star Trek: Enterprise episode, "Unexpected", where it was supposed to represent a more than a century-old design prior to the K't'inga design. In both cases, its use constituted an anachronism in canon.
- Star Trek: The Official Starships Collection, issue 7, 2013
- Klingon K'tinga-class Cruiser at StarshipModeler.com – includes over eighty close-up images of the actual filming model from Star Trek: The Motion Picture and Star Trek VI: The Undiscovered Country
- The Everlasting Battlecruiser at Ex Astris Scientia – a comprehensive look at the lineage of the legendary Klingon battle cruiser