Well, we have to qualify Hamlet as a hero...he is a tragic hero. A traditional hero would have saved the girl and not leave the land strewn with bodies...
Even so, Hamlet, the character, meets Joseph Campbell's definition of a hero. Here is a (very) basic overview of the steps of a hero's journey (as defined by Campbell):
1. Quest: The hero realizes he has special duties or responsibilities. Hamlet, of course, realizes it is his duty, and his alone, to avenge his father's murder by killing Claudius.
2. Discovery: The hero discovers his strength and weaknesses. For Hamlet, his strength and his weakness is one in the same: his intellect. He can create complicated scenarios and look at situations from every possible angle, yet his thinking also hold him back from action.
This weakness is most apparent in Act 3, when Hamlet comes upon Claudius in prayer. It would have been the perfect opportunity to take his revenge, yet Hamlet stops and reasons. He muses:
O, this is hire and salary, not revenge.
He 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?
But in our circumstance and course of thought,
'Tis heavy with him: and am I then revenged,
To take him in the purging of his soul,
When he is fit and season'd for his passage?
Ironically, Claudius had NOT been in confession. Thus it would have been the ideal opportunity yet his intellect prohibits action.
3. Sacrifice: the hero loses something dear to him. In this case, the dear thing Hamlet loses is Ophelia. He is distraught over her death and after losing her, the tragedy picks up speed.
4. Conquest: the hero overcomes death or his fear of mortality. Here, of course, it is the latter. But, in my opinion, when Hamlet is on the verge of death and he says, in awe, "O I die Horatio!" he seems genuinely surprised even as he is accepting his fate. Here, at last, is a situation his intellect is useless against. Hamlet sees there is no way out. He appoints Fortinbras, the man of action, as his successor.
Finally, in Campbell's hierarchy of the heroic journey, there is a messenger who will tell the tale of the hero. The only person of the inner circle alive is Horatio and his close bond with his friend assures that the hero's story will stay alive.