Admiral Marcus sent Kirk and his crew out on a mission to kill the escaped John Harrison/Khan; however, Kirk spared Harrison's life, who revealed in the second half of the movie that Marcus himself has a hidden agenda. So the specific question is: why didn't Marcus go after Khan himself? After Khan's attack on the Starfleet council, Marcus had to publicly respond to the threat, but why take the risk of exposing his conspiracy by sending Kirk, who was not part of it? Marcus had the USS Vengeance at his disposal, which was more than up to the task of going to Kronos and firing the torpedoes; and, if not himself, Marcus could have ordered one of his trusted crew members to do the job, so there would be no risk that his secret plan came to light.
There are several considerations. First of all, Khan's attack demanded an immediate retaliation, and an admiral cannot lead such a mission himself (as they generally do not have command of a ship—Marcus commands the entire fleet). So Marcus had to send someone else, and an experienced captain with a state-of-the-art ship would be the obvious choice. There was probably no captain among his henchmen who could command a ship, just lower level crew. He could not use the U.S.S. Vengeance for it either, as the ship was still a highly secretive military project, and revealing it would evoke too many questions on how he obtained it (and especially which regulations he had to break to get the ship built). Kirk and his crew just happened to be around to do the mission (which is frequently the case within the Star Trek universe), and giving Kirk the assignment would be most logical, as he has identified Khan during the attack, has been fully briefed on the situation, and saved most of the admirals during the attack. Marcus probably hoped that Kirk's desire for revenge would cause him not to ask questions, and simply carry out the mission by killing Khan. He had not foreseen that Kirk would disobey his orders—realizing at Spock's suggestion that they were unjust—and discover the 72 people inside the torpedoes.
But perhaps more important is that Marcus is convinced the growing tensions between Starfleet and Klingons will inevitably lead to war. It is the reason why he sought out Khan and forced him to create a more powerful ship and weapons. Now that he has them, all that he wants is the war to begin. With Khan having escaped to Kronos, he can kill two birds with one stone by eliminating Khan there, and at the same time causing an incident with the Klingons that will lead to war. As Khan mentions to Kirk:
He sent you to use those weapons, to fire my torpedoes on an unsuspecting planet, and then he purposely crippled your ship in enemy space, leading to one inevitable outcome: the Klingons would come searching for whoever was responsible, and you would have no chance of escape. Marcus would finally have the war he talked about, the war he always wanted.
So sending Kirk was not an issue, as Marcus did not expect him and the Enterprise to survive the mission anyway. In fact, when he noticed that the Enterprise was stranded near Kronos, Marcus intercepted them with the U.S.S. Vengeance
to find out what Kirk had learned. He then decided to destroy the Enterprise himself and blame the Klingons, as a justification for his war. He simply had not counted on Kirk's commanding skills, Scotty having discovered, stowing away on and sabotaging the Vengeance
and Khan's superior intellect and strategic skill.