Admiral Alexander Marcus was the head of Starfleet in the mid-23rd century, and a member of Section 31. He began a conspiracy to start a war with the Klingons using the 20th century genetically-enhanced Human criminal Khan. He was the father of Carol Marcus.
Alexander Marcus was the head of Starfleet. At some point, Marcus had a daughter named Carol with a woman who had the maiden name Wallace. He was also the mentor of Christopher Pike and inspired him to join Starfleet. Having been raised by him, Carol believed (ultimately incorrectly) that her father was incapable of killing innocents. (Star Trek Into Darkness)
Revival of Khan
Following the destruction of Vulcan in 2258, (Star Trek) Marcus began seeking ways to militarize and better defend the Federation. In particular, he considered war with the Klingon Empire as inevitable. Section 31 discovered the SS Botany Bay which led to Marcus reviving Khan Noonien Singh and recruiting him under the name "John Harrison" to design weapons and ships.
However, Khan resented Marcus' control over him and the use of his fellow surviving Augments as hostages, and attempted to smuggle them out in the experimental photon torpedoes he created. He was discovered, and Khan was forced to flee alone, beginning a one-man war on the Federation as he presumed Marcus killed his crew.
When Captain James T. Kirk violated the Prime Directive while on Nibiru, the matter was brought to Marcus' attention who, as punishment, confiscated the USS Enterprise from him and sent him back to Starfleet Academy. However, Pike convinced Marcus to alter the punishment for his protégé. Kirk was instead demoted to Pike's first officer.
On stardate 2259.55, Khan coerced Section 31 agent Thomas Harewood into bombing the Kelvin Memorial Archive in London. Marcus received a message from Harewood before he died explaining Harrison threatened him, and the admiral declared a manhunt at a summit in Starfleet Headquarters. Khan, aware protocol dictated a summit like this one, appeared in an attack vehicle and opened fire. However, Khan was forced to flee and Marcus survived, though Pike was killed.
The following morning, Kirk reported Harrison had used the confiscated transwarp beaming formula to escape to Qo'noS. Marcus ordered Kirk to pinpoint and execute Khan with the seventy-two experimental photon torpedoes he had designed.
Once Kirk discovered the truth from Khan, Marcus showed up in the USS Vengeance – a warship designed by Khan – demanding Kirk hand him over. When Kirk attempted to flee to Earth to put Khan on trial and expose Marcus, the Vengeance caught up and crippled the flagship. Marcus' daughter Carol revealed she was on the Enterprise, but the admiral simply had her beamed up and prepared to wipe out all evidence of his scheme. Fortunately, Montgomery Scott had boarded the Vengeance at coordinates given by Khan and deactivated its weapons. Khan and Kirk donned thruster suits and flew over to the ship to commandeer the bridge, during which Khan saved Kirk's life when the display compass in Kirk's helmet went dead leaving Kirk flying blind. Khan found Kirk amid the debris and guided him in.
After taking over the bridge, Kirk, believing that Khan intended to betray them, ordered Scott to stun Khan. Kirk then confronted Marcus over his betrayal of everything the Federation stood for, but Marcus angrily insisted that his actions were justified, and that without him the Federation would be destroyed in what he considered to be an inevitable war with the Klingons, a war he, ironically, was trying to start in the first place. Suddenly, Khan assaulted Kirk and Carol. Marcus briefly escaped to a console near the back of the bridge, but before he could achieve anything, Khan caught up with the admiral and killed him by angrily crushing his skull with his bare hands, prompting a horrified scream from Carol. (Star Trek Into Darkness)
"By now, some of you have heard what happened in London. The target was a Starfleet Data Archive, now it's a damn hole in the ground, forty-two men and women are dead. One hour ago, I received a message from a Starfleet officer who confessed to carrying out this attack, that he was being forced to do it by this man: Commander John Harrison. He's one of our own, and he's the man responsible for this act of savagery. For reasons unknown, John Harrison has just declared a one-man war against Starfleet, and under no circumstances are we to allow this man to escape Federation space. You here tonight represent the senior command of all the vessels in the region, and in the name of those we lost, you will run this bastard down. This is a manhunt, pure and simple, so let's get to work."
- - Alexander Marcus, explaining the London bombing.
"Earth's perimeter sensors have not detected any warp signatures leaving the system, so we know he can't be far. You will park your ships in a blockade formation then deploy search vehicles and landing parties to run down every lead. This man has shown willingness to kill innocent people, so the rules of engagement are simple. If you come across this man and fear for your life or the lives of those nearby, you are authorized to use deadly force on sight."
- - Alexander Marcus, explaining the hunt for John Harrison
"You got something to say Kirk, say it. Tomorrow's too late."
- - Alexander Marcus, to James Kirk about his outburst
"All-out war with the Klingons is inevitable, Mr. Kirk. If you ask me, it's already begun. Since we learned of their existence, the Klingon Empire has conquered and occupied two planets that we know of, fired on our ships half a dozen times. They are coming our way."
- - Alexander Marcus, to James Kirk, about the tensions between the Federation and the Klingons
"Well, shit! You talked to him."
- - Alexander Marcus, upon realizing that James Kirk knows John Harrison's true identity
"That's a hell of an apology, son. But if it's any consolation, I was never gonna spare your crew. Fire when…"
- - Alexander Marcus, after James Kirk offers himself in return for the Enterprise's safety
"You better stop and think about what you're doing, Kirk. You better think about what you did on Qo'noS. You made an incursion onto an enemy planet! You killed a Klingon patrol. Even if you got away without a trace, war is coming. And who's gonna lead us?! YOU?! If I'm not in charge, our entire way of life is decimated! So, you want me off this ship, you better kill me."
- - Alexander Marcus, to James Kirk, with the latter trying to arrest him at gunpoint
Admiral Marcus was played by Peter Weller, who also played John Frederick Paxton in Star Trek: Enterprise. The character was described by producers Damon Lindelof and Bryan Burk as being their attempt to incorporate a trope of Hong Kong cinema into the film, in which the hero (Kirk) and villain (Khan) must temporarily set aside their differences. 
Weller compared Marcus to General Curtis LeMay, the Chief of Staff of the United States Air Force in the 1960s, and a proponent of nuclear war. LeMay "had eighteen nukes stored on his own that he didn't tell the NSC or President Kennedy about – because he didn't deem them knowledgeable enough about war. That is unbelievable arrogance! So, Admiral Marcus is basically like LeMay."  However, he opined that Marcus actually has a conscience because he wants to undo his mistake, but what "makes him bad, from a moralistic view, is that he's willing to sacrifice Kirk and the Enterprise to put this thing back in its shell." 
Weller also disliked that people thought of Marcus as a "bad guy", stating, "Everything he says is true: the Klingons are coming, they do need Khan, and that's that. It's just that he’s going to sacrifice the entire Enterprise to get the job done, because the Enterprise started to believe Khan. But if the Enterprise had not believed Khan and had done what Marcus said, then there'd be no movie, and everything would be cool. But the great writing in this is that the Enterprise wakes the dude up and listens to his game, and then everything goes to crap. But that's the Enterprise's hubris. That's them. They screwed up, not Marcus. Anyway, sorry to go off there. I just hate that." 
Weller added he liked the character's name because it reminded him of Alexander the Great and Marcus Antonius.  The Washington Examiner found parallels between Marcus and Dick Cheney, to which Simon Pegg responded "Absolutely, he's definitely a Republican." 
The IDW comic series Star Trek: Countdown to Darkness, set in the alternate reality, reveals that Marcus was the first officer of a previous starship Enterprise (β) which existed subsequent to Enterprise NX-01, but prior to the new USS Enterprise and was a friend to Captain Robert April (β).