What Hamlet actually says in Shakespeare's play of the same name is:
A villain kills my father, and for that,
I, his sole son, do this same villain send
To Heaven. (Act 3.4.76-78)
There is no mention of Hamlet being afraid of death or hell, here. He mentions his father's state, a state similar to purgatory during which his sins must be burnt and purged, and he compares this to sending Claudius to heaven by killing him while he's confessing his sins. But Hamlet doesn't talk about his own fate.
Furthermore, Hamlet's famous "To be or not to be" speech (Act 3.1) is more of a rumination or contemplation on existence, an intellectual process being revealed, than it is a speech about Hamlet being afraid of death. To be is to exist. Hamlet is reasoning out the question of whether existence is worth the trouble, when one considers all of its hardships and trials. He concludes that if one were absolutely sure about what waited on the other side of death, existence would not be worth the trouble. But since the afterlife is an unknown, one might as well continue existing.
Basically, I suggest Hamlet is not afraid of death. In fact, he suffers from depression or melancholy most of the play and he feels that death might be welcome. The main things that Hamlet is afraid of are acting before his actions are completely and fully thought out, and killing an innocent man.
There are a variety of places that describe or at least explicate Hamlet's fear of death, one of them is described in the previous answer where he talks about the undiscovered country from which no one returns to tell us about it. Another, in the same speech, is his discussion of ending one's life to enter a state of sleep but "perchance to dream." If his father's description of his current woes in what we can presume is purgatory, Hamlet has many reasons not to want to off himself as it is a mortal sin. Because if he does, and he doesn't just sleep but dreams in some way, he may very well be having some very, very bad dreams.
During his father's visitation, he says that he isn't allowed to tell about how bad his current trials are, but if he did, it would do more than curl your hair and shake in your boots. Add these scary things to Hamlet's natural inability to be decisive and you have a recipe that leaves him terrified of suicide, murder and death.
To me, a good place to see that Hamlet is afraid of death is in the "to be or not to be" soliloquy. In that speech, he talks about wanting to kill himself because life is so hard. But then he decides not to. The main reason why he decides not to is that he is afraid of death.
He says that death is an undiscovered country and that no one knows what will happen to us when we die. Because of that, it is a bit scary and that is why he does not want to kill himself.