In Hamlet, why doesn't Hamlet kill Claudius in act 3 when he has the chance?

You are of course refering to Act III scene 3. This is when Claudius is stricken by his guilt after seeing The Mousetrap play especially put on for him by Hamlet, and goes to pray to God. Hamlet passes him by on his way to his mother's chamber and has the perfect opportunity to kill him. Note what he says to himself as he surveys the praying Claudius:

Now might I do it pat, now he is praying,

And now I'll do't, and so he goes to Heaven,

And so am I reveng'd: that would be scann'd,

A villain kills my father, and for that

I his sole son, do this same villain send

To Heaven.

The belief in Elizabethan times was that if you were killed whilst praying you would go straight to heaven and bypass the fires of Purgatory, where your sins during life were dealt with. Hamlet therefore chooses not to kill Claudius because he wants Claudius to suffer for his sins and actions during his life, rather than merely kill him and give him a ticket straight to heaven. He does not want to kill Claudius when "he is fit and season'd for his passage" as he says in the same scene. That would be no revenge whatsoever, especially as his own father's ghost is still languishing in purgatory as he tells Hamlet.

Why didn't Hamlet kill Claudius at the end of act 3, scene 3?

Hamlet tells himself that he can't kill the King while he is praying. But the real reason is probably that Hamlet can never make up his mind. Coleridge said that Hamlet "thinks too much." Hamlet sees too many sides to any question. When he does act, it is always on impulse, before he has had a chance to think. For example, he acts very courageously and decisively when his ship bound for England is attacked by pirates. Coleridge's diagnosis of Hamlet's character is probably the best that has ever been attempted. The answer by rishakespeare at this link below gives another perspective on Hamlet's actions. See what he says about "a harsh reality of an immoral world with his idealistic Christian reality."

http://www.enotes.com/homework-help/soliloquies-hamlet-385630

Why didn't Hamlet kill Claudius at the end of act 3, scene 3?

Hamlet doesn't kill Claudius at this point because he believes that Claudius is praying. He says that killing the king NOW would be "hire and salary, not revenge!" He simply cannot send Claudius to heaven, where he would surely go were he killed just after praying and purging his sins. He thinks that would not avenge his father's murder, because Claudius killed Old Hamlet without giving him the opportunity to pray, and therefore, Old Hamlet must spend time in hell. Hamlet thinks killing Claudius when he is fit for heaven would be like paying Claudius for the murder of his father. That simply would not do.

The real irony is that Hamlet does not realize that his revenge could have been complete if he had actually killed the king then and there, because Claudius was not really praying. Oh, sure, he was on his knees, but Claudius says that he knows he cannot be forgiven for the murder unless he truly repents, and repentance would mean giving up his crown and queen. So when Claudius says "My words fly up, my thoughts remain below./ Words without thoughts never to heaven go," he is really admitting that he knows he isn't forgiven. If only Hamlet had known, then Polonius, Ophelia, Rosencrantz and Guildenstern, Leartes, Gertrude and Hamlet all could have survived.

But then, the story wouldn't be a tragedy :-)

Why didn't Hamlet kill Claudius at the end of act 3, scene 3?

Hamlet doesn't kill Claudius at that moment because he thinks that Claudius is praying. Hamlet says that killing Claudius now would, "send [this same villain] To heaven” instead of Hell where he belongs. Hamlet vows to kill Cladius as Cladius had killed his father, and not send him to heaven.

In Hamlet, why doesn't Hamlet kill Claudius in Act 3, Scene 3's "prayer scene"?

As soon as Polonius exits the scene, Claudius laments about his inability to pray and the guilt he is experiencing after murdering his brother. As Claudius proceeds to contemplate his sin and wonder whether or not he can ask God for forgiveness, he kneels down as Hamlet silently enters the room. Knowing that he has a perfect opportunity to kill King Claudius, Hamlet draws his sword. However, Hamlet hesitates and says,

"And so he goes to heaven. And so am I revenged.—That would be scanned. A villain kills my father, and, for that, I, his sole son, do this same villain send To heaven. Oh, this is hire and salary, not revenge" (Shakespeare, 3.3.75-80).

Hamlet refuses to kill Claudius while praying, because he believes that Claudius's spirit will go to heaven since he is in the process of asking God for forgiveness. Hamlet knows that his father never had a chance to repent for his numerous sins before Claudius murdered him and does not think that sending Claudius's soul to heaven would be adequate revenge. Essentially, killing Claudius is not enough for Hamlet, who wishes to send Claudius's soul to hell for eternity. Hamlet hopes to kill his uncle while he is excessively drinking alcohol, gambling, or "in th' incestuous pleasure of his bed" in order to ensure that his soul will suffer in hell.

Why does Hamlet hesitate to kill Claudius in act 3, scene 3 and when does he decide to do it?

In Hamlet, act 3, scene 3, Hamlet sees Claudius praying and considers killing him immediately. Claudius is on his knees, unwary and defenseless, meaning that it would be easy to do so. However, Hamlet hesitates because Claudius is in the process of confessing his sins and cleansing his soul. Hamlet thinks that if he kills the king now, Claudius will go straight to heaven, since he has no unabsolved sins on his conscience.

Hamlet reflects that to kill Claudius at such a time would essentially be to do him a favor rather than to take revenge. Claudius showed no consideration for Hamlet's father when he murdered him "With all his crimes broad blown, as flush as May." Hamlet must therefore ensure that he kills Claudius

When he is drunk asleep, or in his rage,
Or in the incestuous pleasure of his bed;
At gaming, swearing, or about some act
That has no relish of salvation in't.

If he does so, then Claudius will be in the midst of his sins at the point of death and will go to hell.

It is difficult to discern how serious Hamlet is in offering this reason for refusing to kill Claudius at prayer. It may be yet another excuse for his delay or for his chivalrous unwillingness to stab a defenseless man in the back. From a Christian perspective, however, Hamlet's reasoning is understandable: he has recently received proof of the afterlife and of what happens to those who die with their sins unabsolved, in the form of his father's ghost.

Why doesn't Hamlet take action against Claudius in Act 3, Scene 3?

At this point in the play, Hamlet has witnessed Claudius's reaction to the play, "The Mousetrap," and knows he is guilty of murder. When he sees Claudius at his prayers, all alone and vulnerable, not expecting an attack and not in a position to defend himself, he realizes that here is the ideal moment to avenge his father.

Yet Hamlet does not act. His reasoning for this hesitation is that his father died without having said his prayers. His father was therefore not in a state of grace—his sins were not forgiven, which is why he was doomed to walk the earth as a ghost. If Claudius dies in prayer, he will go straight to heaven. That, to Hamlet's mind, is not an equal retribution.

Therefore, once again, Hamlet hesitates.

In a bit of dramatic irony, we as the audience know that Claudius is incapable repenting of the murder: he is glad he did it, because he likes the power he has gained as a result. Therefore, his soul isn't really clean in the eyes of God—but Hamlet doesn't know this.

Why doesn't Hamlet take action against Claudius in Act 3, Scene 3?

In Act 3.3 of Shakespeare's Hamlet, Hamlet tells the reader why he doesn't kill Claudius and achieve his revenge:

Now might I do it pat, now 'a is a-praying,

And now I'll do't--an so 'a goes to heaven,

And so am I revenged.  That would be scanned.

A villain kills my father, and for that,

I, his sole son, do this same villain send

To heaven.  (Act 3.3.73-78)

Hamlet assumes Claudius is confessing his sins.  According to Hamlet's Catholic beliefs, Claudius' soul would be wiped clean and he would be forgiven and if killed, would go straight to heaven.  Claudius is not confessing his sins, but Hamlet assumes he is.  Thus, he chooses not to kill him and send him to heaven.

This, of course, is probably the climax of the play.  When Hamlet makes this decision he is playing God--dealing with human salvation, not revenge.  Salvation is God's business, not Hamlet's.  Because Hamlet walks away, innocent people die, including Hamlet himself. 

Why does Hamlet decide not to kill Claudius in Act 3, Scene 3? Is he procrastinating?

This scene occurs after Claudius has seen the play which essentially reenacts his murder of his brother, Hamlet’s father. Claudius is trying to pray, acknowledging his crime before God, but not asking for forgiveness. He says he cannot ask for forgiveness:

'Forgive me my foul murder'?
That cannot be; since I am still possess'd
Of those effects for which I did the murder,
My crown, mine own ambition and my queen.

Since he is still in possession of those things which he gained by the murder, he is unwilling to actually ask for forgiveness.

When Hamlet walks onstage, he is unseen by Claudius, who is still kneeling. Hamlet doesn’t hear him and does not know that Claudius is conflicted. To Hamlet, the sight of Claudius praying means that if he kills Claudius now, Claudius will be absolved of his crime and go to heaven. The idea of Claudius going to heaven is unthinkable for Hamlet:

A villain kills my father; and for that,
I, his sole son, do this same villain send
To heaven.

The above passage reveals the bitter irony Hamlet feels at the thought of sending his father’s killer to heaven. So he resolves to kill Claudius some other time:

When he is drunk asleep, or in his rage,
Or in the incestuous pleasure of his bed;
At gaming, swearing, or about some act
That has no relish of salvation in't;

A further irony in the situation is that Hamlet does not realize that Claudius has not asked for forgiveness. Perhaps if he had known this, he might have carried out his revenge.

Although Hamlet sounds earnest in this scene, we must keep in mind that Shakespeare doesn’t always spell things out directly for his reader. Is it really reasonable to presume that it matters when Hamlet kills Claudius? That implies that Hamlet, rather than God, has control over whether or not Claudius goes to heaven. Looking at it this way, it is easy to think that Hamlet may be rationalizing, just looking for a reason not to do it.

In act 3, scene 3 of Hamlet, why doesn't Hamlet kill Claudius?

In Act III, Scene 3, Hamlet has quietly made his way to the door outside Claudius's chamber. He watches as the king kneels to pray, and recognizes that this would in a way be an ideal time to kill him and be done with it. He decides not to do so, though, out of the possibility that, if caught in prayer, the king might go to heaven, which would defeat the purpose of trying to avenge his father's death in the first place. Instead, he says, he will try to catch the king 

When he is drunk asleep; or in his rage; 
Or in the incestuous pleasure of his bed; 
At game, a-swearing, or about some act 
That has no relish of salvation in't...

This will guarantee that Claudius will not make it to heaven, and that his soul will be as "damn'd and black as hell, whereto it goes." Only in this way will his father's murder truly be avenged. Hamlet does not just want to destroy the king's body but his soul as well.