(covers information from several alternate timelines)
The ship, which was considered holy, belonged to T'Kuvma's father. Upon his death, when T'Kuvma was a child, it lay abandoned for children to play in until T'Kuvma rediscovered it and vowed to restore the honor of his House.
In 2256, the Sarcophagus was encountered by the USS Shenzhou at the edge of Federation space, six light years from Gamma Hydra. The vessel later instigated the Battle of the Binary Stars, by firing on the Shenzhou, but was disabled by a bomb afterward, and T'Kuvma was killed by a Starfleet boarding party. (DIS: "The Vulcan Hello", "Battle at the Binary Stars")
The vessel was left stranded in the binary system for six months afterwards, with dwindling food supplies and no way to repair or replace its dilithium processing unit. Kol of the House of Kor eventually arrived and bought the allegiance of the Sarcophagus' crew by bribing them with food, before claiming the vessel as his own and leaving Voq to die on the derelict Shenzhou. (DIS: "The Butcher's Knife Cares Not for the Lamb's Cry")
The Sarcophagus was used by Kol as his personal flagship, directing the war effort from its bridge. When an invitation was received from the planet Pahvo in 2257, Kol ordered the ship to set a course to that destination. (DIS: "Si Vis Pacem, Para Bellum")
The Sarcophagus arrived at Pahvo in a cloaked state, and Kol prepared to destroy the planet but was forced to engage the USS Discovery when the Federation starship appeared in orbit. The Klingon vessel decloaked, allowing the Discovery to use the ship's momentary vulnerability to beam a boarding party aboard. When the Discovery then utilized its spore drive to outmaneuver the Klingon vessel, the Sarcophagus was recloaked, under Kol's orders. However, this was exactly what the crew of the Starfleet ship wanted, as it enabled them to use sensors covertly planted aboard to develop an algorithm to detect the imperfections in the Klingon cloaking technology. With the algorithm completed, the Discovery was able to easily locate and destroy the Ship of the Dead. (DIS: "Into the Forest I Go")
Inside the mycelial network in 2257 of the mirror universe, Paul Stamets and his mirror-universe counterpart briefly viewed a holographic display of the Sarcophagus while attempting to map their way out of the network. The display was overlaid with the coordinates of spore jumps the Discovery had made, but the graphic of the ship soon disappeared, leaving only the view of the coordinates, which the prime-universe Stamets then deactivated. (DIS: "Vaulting Ambition")
The vessel was significantly larger than a Federation Walker-class starship and was covered with thousands of hollow, ornamental, metallic pods that were tightly interlocked to form a symbolic armor. These caskets contained a "Black Fleet" of remains of warriors, dating back as far as thousands of years. T'Kuvma had also devised a way to emit a form of stealth mode described as a cloaking screen. The ship could generate a multitude of tractor beams from its "neck" section. It was armed with directed energy weapons and torpedoes, and possessed enough firepower to destroy the surface of a habitable planet. (DIS: "The Vulcan Hello")
The viewscreen display of the vessel aboard USS Shenzhou identified this vessel as an "Unknown Klingon Command Vessel". The series' press kit identified it as "The Sarcophagus Ship", while other background sources identified it under the similarly titled "Klingon 'sarcophagus ship'".  
A feature on the ship in SFX magazine (issue 292, p. 83) described the ship as two hundred years old, and three times the size of its Federation counterparts. It was said to have no consoles, because its crew interfaced with their computers directly via their ornate silver masks.
Flooring from the Sarcophagus' bridge was synthetically aged and then reused, combined with various other repurposed set pieces, in the creation of Sarek's camp in "The Wolf Inside". (AT: "The Wolf Inside") The set of the Sarcophagus' graveyard chamber was likewise reused for production on "Will You Take My Hand?", to create the set of a cabaret in an Orion outpost on Qo'noS. On that occasion, the graveyard chamber set was combined with a reuse of a set that had served, earlier in the show's first season, to represent the ISS Charon's throne room. ("Designing Discovery", DIS Season 1 DVD & Blu-ray special features)
In the Star Trek Online short story "Step Right Up", a disreputable Klingon archaeologist named Tortoth claimed to have recovered the ship's schematics from the wreckage of its final battle at Pahvo, and revealed a rebuilt Sarcophagus – equipped with modern (25th century) technology – at a press conference on Risa, where he invited viewers to place orders for a Sarcophagus of their own.  The blog post coincided the ship's release in the "Discovery Lock Box" in January 2018, designated as the "Sarcophagus Dreadnought Carrier". The Sarcophagus also appears in a "Task Force Operation" – a multi-player mission – that recreates the Battle of the Binary Stars.