What was Khan’s problem?

Because he blamed Kirk for the bad stuff that had happened to him and his posse.

He and Kirk have a history, and Khan also has a history going back centuries. You would need to watch the old Star Trek Series episode "Space Seed," to see how he and Kirk first met.

In their timeline, Khan was one of the last great tyrants of Earth, living during the early 21st century. It was a time when they had humans who had been genetically modified to be super-humans (they were called Augments). The trouble with Augments was the fact that the methods used to make them caused them to go nuts and become dangerous sociopaths. Khan was the worst of them, because he was cunning, highly intelligent, and a military genius. He apparently had taken a bunch of his Augmented buddies and had secretly escaped Earth during one of the last big wars between regular humans and Augments. The entire idea was to stay out in space long enough to "outlive" the war, and either colonize another planet to start a new empire, or wait until they could get back to earth and re-conquer it. So everyone on the ship was cryogenically frozen to extend their lifespans and "wait" it out until someone came along to wake them up.

That's how Kirk's crew found the Botany Bay (the ship's name) floating in space centuries later, and woke up Khan's crew, having no idea what kind of Pandora's Box they'd opened. It took some time to figure out what a monster Khan was, and they almost lost the Enterprise and almost got Kirk killed because of this guy. Kirk and Starfleet punished Khan by giving him exactly what he wanted: banished him and his Augment buddies to an earth-like planet that was uninhabited. They gave him a hellish "kingdom" to rule, in a manner of speaking but he had no advanced tech to help him set up a home there, and no starships available, so he and his people could never leave.

We don't find out until decades later, in this film, that life was a disaster for these people. One of the nearby planets exploded, changing the orbit of the one Khan's people were living on, and making the environment almost unlivable. Even worse, Starfleet had kept information about this planet classified, so the officers who ended up on this planet by accident had no idea what kind of trap they'd walked into.

Khan had also harbored an extreme grudge against Captain Kirk, due to thwarting his plans for intergalactic domination decades ago, and exiling him on that planet that turned into a hellhole. This entire movie was him taking advantage of the situation to get revenge on his most hated enemy, and Ricardo Montalban did a brilliant job with the role.

Because, on earth, 200 years ago, he was a prince....

And Shitner ruined that for him.

Why was he all wrathy?