KHAN: You are in a position to demand nothing, sir. I, on the other hand, am in a position to grant...nothing. What you see is all that remains of the ship's company and crew of the Botany Bay, marooned here fifteen years ago by Captain James T. Kirk.
Khan pushes this as the reason he wants to get even with Kirk, and it's a good one. But it's not the only one. There's his wife, for starters, which is the real purpose of the exercise and which slithers out in the odd line or two.
CHEKOV: You lie! On Ceti Alpha Five there was life, a fair chance!
KHAN: This is Ceti Alpha Five! Ceti Alpha Six exploded six months after we were left here. The shock shifted the orbit of this planet and everything was laid waste. Admiral Kirk never bothered to check on our progress. It was only the fact of my genetically engineered intellect that enabled us to survive!
Yeah, that would make us cranky too. We're not saying that what Khan does is right…but he's definitely got a legitimate beef.
JOACHIM: We're all with you, sir, but consider this. We are free. We have a ship and the means to go where we will. We have escaped permanent exile on Ceti Alpha Five. You have proved your superior intellect, and defeated the plans of Admiral Kirk. You do not need to defeat him again.
KHAN: He tasks me. He tasks me and I shall have him. I'll chase him round the moons of Nibia and round the Antares maelstrom and round Perdition's flames before I give him up.
First of all, shout-out to the Moby-Dick quote here, which Meyer set up by showing us a copy of the book itself on the Botany Bay. More important, it reveals another possible reason for revenge beyond the dead wife thing and the exile thing. It's plain old-fashioned ego.
Kirk beat Khan, and Khan is not a man who likes to be beaten. So he's coming back for the grudge match, just to show Kirk who really has the biggest stones in the universe.
KHAN: Ah, Kirk, my old friend, do you know the Klingon proverb that tells us "Revenge is a dish that is best served cold?" It is very cold...in space.
This is a great line. It's intended to convey the notion that the best revenge is one you take your time with: where you think it through and plan it out with preparation and care instead of rushing off half-cocked in a blind rage. The odd thing is, that doesn't sound very Klingon, since they're a shoot-first-and-ask-questions-later kind of species.
The phrase itself can be traced back centuries, but the first actual appearance of it was in the 1885 play Prince Bismark, which reads "we did not act like drunken demagogues, but like responsible politicians who saw the wisdom of eating cold the dish of their revenge." (Source)
It's also been attributed to the Mafia, and a number of other sources. But if we're being frank, it was Khan that really brought the statement into the popular front… so much so that Quentin Tarantino cited it as an "old Klingon proverb" at the start of his revenge epic Kill Bill.
KHAN: You still remember, Admiral. I cannot help but be touched. Of course, I remember you.
KIRK: What is the meaning of this attack? Where is the crew of the Reliant?
KHAN: Surely I have made my meaning plain. I mean to avenge myself upon you, Admiral. I've deprived your ship of power and when I swing round I mean to deprive you of your life. But I wanted you to know first who it was who had beaten you.
See, there's the ego again. Had Khan just opened fire, the Enterprise would have gone down in a hail of blazing visual effects, and vengeance would be satisfied. But Khan just has to rub Kirk's nose in it, and in the process undoes the immense tactical advantage he had. Way to go, Captain Narcissist!
KIRK: You were going to kill me, Khan. You're going to have to come down here! You're going to have to come down here!
KHAN: I've done far worse than kill you. I've hurt you. And I wish to go on... hurting you. I shall leave you as you left me. As you left her. Marooned for all eternity in the center of a dead planet...buried alive…buried alive!
Here we get Khan's first, best reason for revenge: his wife, who died on Ceti Alpha Five and who he clearly loved like nobody's business. He rarely allows himself to express that pain, but when it comes out…man does it send chills down the spine.
(Also, KHHHAANNN!!! is still just so epic.)
KIRK: This is Admiral Kirk. We tried it once your way, Khan. Are you game for a rematch? Khan? I'm laughing at the superior intellect.
KHAN: Full impulse power!
JOACHIM: No sir! You have Genesis! You can have whatever you—
KHAN: Full power! Damn you!
And…yeah. This would be a time to walk away ahead of the game. But Khan, being Khan, just isn't able to. His quest for revenge has caught him in the same trap he's laid for Kirk. And we'd be remiss if we didn't mention Khan's resemblance to another vengeance-crazed captain here: a one-legged dude named Ahab.
JOACHIM: Yours... is... the superior...
KHAN: I shall avenge you.
It's a nice thought, but—having already doubled down on the whole revenge thing—it's going to be hard for Khan to get even more revenge-y solely for the sake of poor old sensible Joachim.
KHAN: No Kirk…the game's not over…to the last I will grapple with thee…
A little Ahab quote, just to make sure we're absolutely clear how far Khan is willing to go here.
KHAN: Now... Now… You can't get away... From Hell's heart, I stab at thee! For hate's sake I spit my last breath at thee...
Our final Moby-Dick reference, and it's worth noting that Khan deliberately chooses to quote Ahab's last words as his own. He regrets nothing—and if he had a chance to do it again, he probably wouldn't change a single thing.