and b) what do we, as the audience (or readers) of the playthink about why Hamlet doesn't kill him?  Neither of these may definitively answer the question, but it's a good place to start our exploration of why Hamlet doesn't take this very easy opportunity to kill his hated stepfather.

Let's first explore what Hamlet says about it (because, in true Shakespearean fashion, most or all of a character's emotions and motivations are displayed to the audience by the device of soliloquy, which is really just the character talking to himself.)  This is Act III, scene 3.

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 revenged. That would be scann'd. A villain kills my father; and for that, I, his sole son, do this same villain send To heaven.(80)

...No. Up, sword, and know thou a more horrid hent.(90) 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 Then trip him, that his heels may kick at heaven,(95) And that his soul may be as damn'd and black As hell, whereto it goes.

What's young Hamlet saying here?  He's saying that he could do it "pat" (easily) -- kill his stepfather easily, without too much risk to himself.  But he doesn't -- and the reason he says that he doesn't is that this would be "hire and salary" (III.iii.81) (in other words, he'd be doing Claudius a favor by sending him to heaven when his soul was clean, and therefore enabling this murderer to go to heaven).  Hamlet says that he wants to "take" (86) Claudius when he is sinning, and then he enumerates those sins.  This would mean, according to Hamlet's Catholic beliefs, that Claudius would go to hell, or at least purgatory, for his unforgiven sins.  Hamlet has this foremost in his mind because he has just seen the ghost of his father, who has told him that he is in purgatory (Act I, Scene iv).  Of course, the vengeful son would want at least as bad an afterlife for his father's murderer as for his own father.  This is a neat, doctrinally sound excuse for not killing Claudius -- if you overlook the fact that Christian doctrine expressly forbids killing!  So Hamlet is splitting hairs here -- and we think that he knows it.

So, the audience/reader is left to think of why Hamlet might not be motivated, at that moment, to kill Claudius.  One might be for his own safety; he might be afraid of Claudius's friends (not to mention his own mother's, Claudius's wife) reprisals.  Perhaps he shrinks from killing a king because people might accuse him of the same thing that Claudius did:  of killing a sitting king in order to inherit his throne for himself!  This would certainly give any honest, moral person pause, especially when the murdered person was his own father.  Or, perhaps, we are meant to think that Hamlet is truly a good person, and is only driven towards murder because of circumstances, not because of his nature.  These are all valid interpretations, and there are many others.

Hamlet, in Shakespeare's Hamlet, tells you himself why he hesitates to kill Claudius:

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

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

And so am I revenged.  That would be scanned [looked at or thought about again].

A villain kills my father, and for that,

I, his sole son, do this same villain send

To heaven.

Why, this is hire and salary, not revenge.  (Act 3.4.73-79)

Hamlet's Catholic beliefs suggest that if Claudius dies just after he's confessed, he will be forgiven of all of his sins and he will go to heaven.  Hamlet doesn't want to send Claudius to heaven. 

Hamlet's father, in the form of the Ghost, is lingering in a purgatory-like state, and Hamlet, if he were to kill a king, may go straight to hell.  Thus, Hamlet reasons, Claudius, the evil one of the three, will be the only one of the three to be rewarded with eternal salvation if he kills Claudius while he is confessing. 

Ironically, Claudius, while he is praying, is not really confessing.  He is unwilling to give up the benefits of his sin, and therefore does not confess or repent.  Hamlet could have killed Claudius while Claudius was praying, and order would have been restored in Denmark, presumably.

This makes Hamlet's decision to walk away and to not kill Claudius the climax of the play.  Hamlet is guilty of hubris, attempting to rise above his station in life.  Salvation is God's business, not Hamlet's.  Hamlet is playing God when he attempts to determine another human's eternal salvation.  The terrible loss of so many lives after Hamlet decides not to kill Claudius could have been avoided. 

To a large extent, Hamlet's vacillation is expressive of his double role as both a Christian and a Renaissance prince. He seems constantly torn between these two significant aspects of his personality, so much so that his freedom of action is seriously impaired. Hamlet's existential dilemma is encapsulated in act 3, scene 3, when he spies Claudius kneeling at prayer, begging God for forgiveness for his murder of Hamlet's father:

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 revenged. That would be scann’d:A villain kills my father; and for that,I, his sole son, do this same villain sendTo heaven.

Hamlet's itching to kill Claudius. And this is the perfect opportunity for him to settle scores. He unsheathes his sword, but Hamlet, being Hamlet, changes his mind. His rationale is that killing Claudius while he's at prayer might actually send him to heaven rather than hell, and Hamlet doesn't want to take that chance. Far better to wait until Claudius is doing something depraved like swearing at the gaming table, lying in a drunken stupor, or sleeping with Gertrude.

Clearly, Hamlet wouldn't be thinking this way if Christianity didn't have some effect upon him. He's already indicated how seriously he takes the Word of God in relation to suicide:

That the Everlasting had not fixedHis canon 'gainst self-slaughter (act 1, scene 2).

Also, Hamlet's faith prevents him from completely trusting the Ghost. How does he know it's really the ghost of his father and not some evil spirit?

The spirit that I have seenMay be the devil: and the devil hath powerTo assume a pleasing shape; yea, and perhapsOut of my weakness and my melancholy,As he is very potent with such spirits,Abuses me to damn me: I’ll have groundsMore relative than this: the play's the thingWherein I’ll catch the conscience of the king (act 2, scene 2).

In time, Hamlet comes to see Claudius's treacherous act of murder as part of a bigger picture, a gigantic cosmic drama created by Divine Providence. It is only by seeing his planned revenge as expressive of a higher, more transcendent justice, as opposed to the settling of a personal vendetta, that Hamlet is finally able to reconcile himself to the deed.

The most obvious place where Hamlet hesitates to kill Claudius comes in Act 3, Scene 3, where Claudius is alone, defenseless, kneeling at prayer, and unaware of Hamlet's presence. Hamlet actually draws his sword and says to himself, "Now might I do it pat, now he is praying; / And now I'll do't" (3.3.73-4). But characteristically he has second thoughts. He always has trouble making up his mind to take action, although he can act vigorously at critical moments when he doesn't have time to think, as when he kills Polonius, when he boards the pirate ship, when he fights with Laertes at Ophelia's grave site, and when he fiinally does kill Claudius in the last act. Hamlet tells himself that this is not the right moment because Claudius might be in a state of grace and go to heaven, whereas Hamlet's father was sound asleep and Claudius "...took my father grossly, full of bread, / With all his crimes broad blown, as flush as May; / And how his audit stands who knows save heaven?" (3.3.80-82) Hamlet decides to wait and kill the king "When he is drunk asleep, or in his rage, / Or in th' incestuous pleasure of his bed; / At gaming, a-swearing, or about some act / That has no relish of salvation in't" (3.3.88-92). This may be only partially an excuse for further procrastination.

Precisely because he is praying. Hamlet thinks that if he kills Claudius while he is praying, Claudius will go straight to heaven. He doesn't want him to go to heaven, so he holds back.