In Shakespeare's Hamlet, Claudius and Hamlet are in conflict from the opening scene to the final scene, including Act 2. Hamlet clearly has the advantage by the close of Act 2--at least as much as anyone could have an advantage over a king.
Hamlet has the advantage for two reasons. First, Claudius thinks Hamlet is "mad" and is preoccupied with finding out the specifics about his madness. Hamlet is just acting, of course, but Claudius doesn't know that. Hamlet is really in the process of planning revenge while Claudius wastes time trying to find out what is making Hamlet act the way he is.
Second, Hamlet knows Claudius killed King Hamlet. Hamlet needs corroboration and further proof that the Ghost is telling the truth about his father's murder, but he is in the process of getting that proof. This is a strong advantage, since Claudius doesn't know Hamlet knows. Politically and strategically, this gives Hamlet the upper hand, so to speak.
The arrival of the players adds to Hamlet's advantage. They provide the means for Hamlet to corroborate the Ghost's story--the play-within-the-play, which will reveal Claudius's guilt with certainty. At the end of Act 2, Claudius has no way of knowing that Hamlet has learned of Claudius's murderous act by supernatural revelation. This, as well as Claudius's preoccupation with Hamlet's behavior, clearly gives Hamlet the advantage.
Well, since Hamlet and Claudius never even speak to each other in Act II, I'm guessing you meant Polonius? The two of them do, indeed, seem to have a bit of a conflict toward the end of this act.
Polonius is convinced Hamlet is mad with love for his daughter, and Hamlet is clearly playing that up in this scene. He calls Polonius a fish-monger, tells him to take care that his daughter doesn't get pregnant...among other disrespectful things. They're interrupted by Rosencrantz and Guildenstern, then the players, and the two of them begin again. They have a disagreement about the speech the player is giving. Polonius says it's too long (isn't that ironic) and Hamlet says, in essence, play on!
Their final disagreement comes when Hamlet asks Polonius to show the players to their rooms and treat them well. Polonius says he will treat them as they deserve. Hamlet gets angry at that and says,
"Use every man after his desert and who shall scape whipping?"
As the last Player leaves, Hamlet says, "...look you mock him not." Clearly he does care for the man despite this verbal sparring, even if it's only because Hamlet cares for his daughter. Hamlet appears to have the advatage, because Polonius thinks he's mad and treats him rather delicately because of it. The next time they see one another, Hamlet mocks Polonius even more, and this time publicly; and Polonius just takes it.