Hamlet is not a hero in the conventional sense. If you compare him to Braveheart, for instance, you might find that he comes up short. He doesn't lead armies of men against an oppressor. He doesn't truly sacrifice himself for a cause greater than himself. His actions bring about the death of his girlfriend's father and her madness. But Hamlet has much to admire: he is brave enough to face the ghost in Act 1; he is clever enough to manuever around Claudius' spies Rosencrantz and Guildenstern in Act 2; he is imaginative enough to stage a play to determine Claudius' guilt in Act 3; he is crafty enough to send Rosencrantz and Guildenstern to their deaths in Act 4; and in Act 5, he is skilled enough to evade Laertes poison sword during the fencing match until Laertes strikes dishonorably.
Perhaps, though, Hamlet fits the definition of a tragic hero more clearly. Aristotle defined a tragic hero as a person of noble birth, having great potential, but through some flaw or error in judgment begins a series of actions that end in a catastrophe, a catastrophe that he in some way causes. As Ophelia claims Hamlet was the "rose of the fair state." You can find more descriptions of Hamlet's enormous potential in Ophelia's soliloquy that ends Act 3, scene 1. Because of his mission to avenge his father's death, Hamlet loses everything, and suffers far more than he deserves.
If you need a scene showing Hamlet as a hero in Shakespeare's Hamlet, look at Act V:ii. Starting in line 292, the following occurs:
The queen swoons, and screams she's been poisoned by drinking the drink that the king had intended for Hamlet.
Hamlet immediately orders the doors be locked, to keep anyone who's guilty from escaping.
Laertes tells Hamlet that the tip of the sword that he used to wound Hamlet was poisoned, that Gertrude is poisoned, and that the king is to blame.
Hamlet immediately attacks the king, wounding him with the poison-laced sword.
The king doesn't immediately die, so Hamlet forces him to drink the poison that killed the queen.
As Hamlet dies, Horatio wants to join him and drink the poison himself. Hamlet orders Horatio to give him the cup and not drink from it.
In the heat of the scene, in the heat of the battle, Hamlet takes control of the situation and attacks immediately. When no doubts exist, here, Hamlet is extremely decisive.
Hamlet is, of course, a tragic hero, but that's another story and I don't think that's what you are asking for.