This isn't really covered in-game (in fact, a lot of fans got upset about changes to his character). That said, you can draw some logical plot lines between the two games.
The main thing is time. When SC2 starts, it has been a few years since the Queen of Blades backstabbed pretty much every last person that was near her at the time. According to SC2, she hasn't been seen in all that time. So Kerrigan is (likely) holed up on Char where practically no one can touch her. Raynor, with an army still recovering from all the crap she pulled, certainly can't.
So as shown in SC2, with no way to fix things and no way to eliminate Kerrigan, he starts spending more time getting his drink on. Rage gets dulled and we get the more bitter and morose Raynor who spends his time guilt-tripping himself over the event that sets her up as the Queen of Blades, Mengsk's betrayal of her on Tarsonis (it even features in a nightmare cinematic). It's not his fault, but it's the starting point, so he fixates on it. Since Mengsk is going out of his way now that he's rebuilt the Dominion to tell the story of Jim Raynor the Terrorist Jackal, it's not out of Jim's way to do something about him instead. Hell, Jim was in love with Kerrigan, so it's revenge from a different direction.
By the time Kerrigan shows herself again, Raynor is fully engaged with the artifact jobs and his momentum against Mengsk to really pin her down. It's probably just an excuse to put off the deed, but it's enough. The one time she might be exposed is when she takes a personal hand on Tyrador VIII, but Jim has his hands full evacuating civilians, Moebius, and cleaning their data trail. A little more time.
Then Valerian comes forward with one way to reverse that one event that haunts him. Of course he jumps for it.
These are easy enough connections to make, and probably just as easy to demonstrate in game, so I don't really understand why they didn't. As such, this answer is relegated to theory, even if it's likely.