The first of Hamlet's three attempts to kill Claudius takes place in act 3, scene 3. Claudius is kneeling down at prayer, seeking forgiveness for the murder of his brother, old King Hamlet. Hamlet spies him from a distance and realizes that with no one around, he has the perfect opportunity to strike and gain revenge for the death of his father.

But Hamlet, being the arch-procrastinator that he is, doesn't go through with it. He figures that if he kills Claudius while he's praying for forgiveness, then there's every chance that his wicked uncle will go to heaven, and that's the last thing that Hamlet wants. He wants Claudius to be punished for all eternity in hell for his wicked misdeeds.

The next opportunity for killing Claudius comes in the very next scene of the play. Hamlet wrongly believes that Claudius is hiding behind a curtain and eavesdropping on a conversation between himself and Gertrude. Hamlet thrusts his sword through the curtain and kills what he initially believes to be Claudius. In actual fact, however, it's the hapless Polonius who was hiding behind the curtain all the while, and it is he who now lies dead.

Hamlet eventually manages to finish off Claudius, but only just before he himself expires. The devious Claudius had arranged it so that Hamlet would be killed by a poisoned-tip sword wielded in a duel by Laertes. Not taking any chances, he's also poisoned a cup of wine that he hopes Hamlet will drink.

Unfortunately for Claudius, his best-laid plans don't work out quite the way he'd expected them to. Not only does his wife, Gertrude, die after drinking from the poisoned cup, but Laertes dies after Hamlet stabs him with the poison-tipped sword. Hamlet is himself poisoned, and before he dies, he angrily runs Claudius through with the poison-tipped sword and forces him to drink from the poisoned cup. At long last, and in his last few moments on earth, Hamlet has finally gained revenge for the death of his father.

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.

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:

HAMLET:
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.