Why does Hamlet hesitate over whether or not he should kill the king, his uncle Claudius?

Several things probably keep Hamlet from "doing the deed."

1.  What Hamlet says is he doesn't want Claudius, the murderer of his father, to have more preparation for death than Claudius allowed King Hamlet.  The Ghost reveals to Hamlet that he died with his sins unconfessed; Hamlet is reluctant to kill Claudius as he is confessing. That would not be the purest form of revenge. The irony is, of course, that Claudius is not actually confessing his sins, so it might have been a perfect time for action.

2.  Hamlet also says he would rather "catch" him doing something else when he chooses to take his revenge.  He knows Claudius is a man of the flesh, and if he catches him in some frivolous or lustful pursuit Claudius will most certainly meet his demise in a sinful condition.

There are several other, more human nature kinds of considerations which may have prompted Hamlet to hold back his hand of revenge when presented with the opportunity.

3.  Hamlet is reluctant throughout the play and has demonstrated his ability to be--at least to some degree--a man of no action.  He is all hyped up for revenge, yet he has to check and re-check and ask a friend (Horatio) to bolster his certainty that the words of the Ghost were not a lie.

4.  Hamlet is not God, and taking a life is a mortal sin according to his religion.  Eternal personal damnation would, I think, give one pause.

5.  Hamlet is as certain as one can be by the time he sees Claudius alone with this opportunity in front of him, yet he has to be wondering how he would live with himself if he killed his uncle and then found out he was wrong.

6.  Revenge is always sweeter, they say, when the person who has done the wrong knows he's being repaid for his crime.  That might have happened in this particular scenario, though it's not likely unless Hamlet told on himself.  The more public end to Claudius, though unplanned, is much more gratifying, for then all knew of his perfidy and crime.

Hamlet could but does not exact his revenge at this moment in the play; this choice determines nearly everyone else's fate, including his own.