Hamlet tries to kill Claudius three times? What are they?
It's an interesting question, but I'm not sure Shakespeare really specifies how many chances Hamlet gets to potentially kill Claudius. Thinking about it, any time when Hamlet is onstage with Claudius, he could potentially pull a knife and just stab him to death - of course, Hamlet knows Claudius is the murderer, we assume (though again, this is difficult to ascertain precisely) from the play scene onward, when he longs to "catch the consciience of the king".
I'd argue Hamlet has a pretty clear chance of assassinating Claudius there and then in the theatre, once he's seen the reaction... though of course, that's not what he says to Horatio he'll do. He's only there to watch Claudius while Claudius watches the play.
I suppose the first real moment that Hamlet has an undoubted chance to kill Claudius is in Act 3, Scene 3:
Now might I do it pat, now he is praying;
And now I'll do't.
But Hamlet talks himself out of it (again). And then after that, any moment at which he sees Claudius he could kill him. So that could include the graveyard scene, the "Where is Polonius?" scene, the fencing match (after which - in the same scene) he does eventually kill him.
There are lots of moments where Hamlet potentially could kill Claudius, though I'm not sure I can give you a definitive number. Key point is, of course, that Hamlet doesn't - not until the end.
There are three times Hamlet tries to kill Claudius. The first time occurs right after the play "The Mousetrap". Claudius is praying, unguarded, and Hamlet has a perfect opportunity to stab him. However, according to Elizabethan belief, killing Claudius while he is in prayer and supposedly confessing his sins, would send him straight to heaven. Hamlet's father was killed while he was sleeping and was not able to confess his sins and therefore, is still in purgatory while his sins burn away. So Hamlet decides to wait until Claudius is doing something sinful and then kill him. The irony is that once Hamlet leaves, Claudius reveals that he is really not confessing and therefore, he would have gone to hell if Hamlet had killed him.
The second time he tries to kill Claudius is a case of mistaken identity. When Hamlet is talking with his mother, he hears someone behind a curtain in his mother's room. Assuming this to be Claudius, he stabs the person behind the curtain and kills him. Unfortunately, it was Polonius who was listening to the conversation between Gertrude and Hamlet and poor Polonius was killed instead of Claudius.
The final time Hamlet tries to kill Claudius, he succeeds. This occurs during the final scene of the play. Unfortunately for Hamlet, he has already been poisoned by Laertes sword and so Hamlet dies soon after killing Claudius.