killing Hamlet's father and by not killing Claudius, when he had a perfect opportunity to do so when Claudius was praying.

He has a tragic mistake (hamartia)—indecision, or "thinking too much"—that ultimately leads to his downfall.

He experiences a moment in the play when he makes an important discovery (anagnorisis). This occurs when Claudius has a guilty reaction to the play-within-a-play that Hamlet arranged to be performed at court, by which Hamlet hoped to "catch the conscience of the king."

Hamlet undergoes a significant "reversal of fortune" (peripeteia) which changes the course of his life when he accidentally kills Polonius.

As the result of his tragic flaw, Hamlet suffers a punishment (nemesis) that he didn't really deserve. Laertes stabs Hamlet with a poisoned fencing sword, which results in Hamlet's death.

Hamlet's misfortune and downfall (his death) elicit pity and fear (catharsis) from the audience.

There are times in the play, however, when Hamlet appears less than heroic.

Hamlet's treatment of Ophelia in act 3, scene 1, the "get thee to a nunnery" scene, is appalling. It's true that Hamlet is grieving about his father's death, and he's feeling manipulated by Claudius, Polonius, and even Ophelia, but that's no excuse for his behavior toward Ophelia.

On his way to England, Hamlet rewrites the letter from Claudius to the King of England from a request for Hamlet's death to a request for the death of Rosencrantz and Guildenstern. Hamlet could have changed the letter to anything he wanted, but he vindictively chose to condemn Rosencrantz and Guildenstern to death for betraying him to Claudius.

Hamlet tries to justify his behavior by saying that Rosencrantz and Guildenstern knew what they were getting into when they agreed to spy on him for Claudius, and they also should have known better that to come between Claudius and himself.

HAMLET. (to Horatio) Why, man, they did make love to this employment!
They are not near my conscience; their defeat
Does by their own insinuation grow.
'tis dangerous when the baser nature comes
Between the pass and fell incensed points
Of mighty opposites.

Hamlet's rationalization for the deaths of his childhood friends rings extremely hollow.

Hamlet seems notably callous about Polonius's death as well. Immediately after killing Polonius, Hamlet returns to berating his mother, completely ignoring Polonius's dead body lying on the floor in the same room.

Hamlet's rationalization for Polonius's death is essentially "wrong place, wrong time" and that it's Polonius's own fault for being such a nosy busybody.

HAMLET. For this same lord,
I do repent; but heaven hath pleas'd it so...
I'll lug the guts into the neighbour room.
...Indeed, this counsellor
Is now most still, most secret, and most grave,
Who was in life a foolish prating knave.

At times, Hamlet exhibits the characteristics of a "non-hero"—lacking virtuous qualities and behaving impulsively and erratically—but for the most part, Hamlet fulfills the definition of a classic tragic hero.

Hamlet did not intend to kill Polonius. If he thought at all in his heightened state of rage and fear, he thought it was Claudius behind the curtain; hence he says, “Sir, I mistook you for your better.”

Hamlet, however, cannot simply be considered a hero because of his despair. What we need to look at is his indecision and his failure to find a way to act either in accord with his father's Ghost's demands for a revenge killing or with his Protestant beliefs that forbid revenge killing.

Of course Hamlet is a hero.  A tragic hero, anyway.  He's basically good: a good son trying to avenge his father (even when his religious beliefs forbid revenge) and to woo an innocent virgin.  However, he, of course, has a tragic flaw the nature of which many contest.  My opinion is that his tragic flaw is inaction.  Quite a mild flaw, I would say, as compared to someone like Macbeth.  Doesn't "vaulted ambition" sound worse to you?

It is possible to make a case against Hamlet being a hero. He acts impulsively and kills people, then bemoans his actions in soliloquies of self-doubt. Admittedly, Polonius was meddling, but one should check curtains before stabbing them willy-nilly.

Hamlet values honesty and loyalty, yet is totally dishonest, sly and sneaky. He has no loyalty for Ophelia and is so cruel to her that, ultimately, she dies by her own confused actions.

Hamlet does have some incredibly heroic qualities, but a little dissension always helps a discussion, I think.

Hamlet is a hero. He is also crippled by the welter of emotions that bombard him after finding out that his father is dead and his mother is married to his uncle.  Upon returning home from university at Wittenberg, he is confronted with the news, delivered by a supernatural form of his father's ghost, that Uncle Claudius is his father's murderer.  This comes on top of Claudius's theft of his birthright, the royal monarchy, passed down by direct bloodline; war with Fortinbras; grieving that death brings to us all. 

In the end, Hamlet did what he said he would do: He found the truth and crushed Claudius's hopes of a kingly life. He then, since his life was the price of vengeance, honorably passed on the crown to Fortinbras with the help of Horatio, his steadfast friend.

In his criticism of Hamlet, renowned critic Harold Bloom writes,

No other single character in the plays, not even Falstaff or Cleopatra, matches Hamlet's infinite reverberations.

Perhaps it is because of these "reverberations," the tremendous scope of Hamlet's character as he echoes so many human traits, that raises the question of his heroism. For, Elsinore is too small a "mousetrap" for one so grand and charismatic as Hamlet, except for the fact that he returns to it voluntarily.  And, thus he is the tragic hero, for like his counterpoint, Fortinbras, Hamlet chooses to avenge his father's honor and sacrifices his life in the heroic effort.

Rightly to be great 
Is not to stir without great argument, 
But greatly to find quarrel in a straw 
When honour's at the stake. How stand I then, 
That have a father kill'd, a mother stain'd,...  
And let all sleep, while to my shame I see 
The imminent death of twenty thousand men...  
Go to their graves like beds, fight for a plot....(4.4.55-64)

Declaring himself in Act V as "Hamlet the Dane," after having defined himself in the graveyard scene as he comes forward to protest Laertes's actions, saying, "Yet have I in me something dangerous," Hamlet unselfishly avenges his father's death against Laertes and Claudius who have plotted against him, bequeathing his kingdom to the "gentle prince," Fortinbras. Indeed, there is a "divinity that shapes our ends" as Hamlet tells Horatio. And, Hamlet, who transcends the range of the human senses, is truly a hero because

1. He is a person of noble stature
2. His actions have lasting results
3. He possesses virtue which proves fatal
4. He has errors in judgment at times
5. He sometimes has distorted perceptions of reality
6. He suffers inwardly and outwardly
7. He elicits pathos from the audience
8. He dies.

Hamlet is not a hero in the play, if we adhere to a conventional definition of a hero as someone who acts with courage and consistent moral principles.

Hamlet has the self-awareness to understand and lament his own lack of courage in avenging his father's death. In his soliloquies, he comments on this cowardice, and he contemplates suicide as a welcome antidote to the problems that beset him. Ironically, he notes that it is cowardice, too, that keeps him from killing himself: he fears what he does not know that might be in the "undiscovered country" of death. Later, he will say of all his thinking and not acting that it is "a thought which, quartered, hath but one part wisdom and three parts coward."

Hamlet is to be commended both for his relentless understanding of his own cowardice and his prudence: he does go to lengths to test whether the ghost is telling the truth. At the same time, cowardice is not a typical attribute of a hero.

More problematic is Hamlet's inconsistent moral principles when it comes to murder: he recoils in dread from killing Claudius, manufacturing excuses not to kill him even when he knows beyond a reasonable doubt that Claudius killed his father. He could, for example, have killed Claudius at his prayers, but he lamely backs off with the rationalization that Claudius would go straight to heaven while his own father suffers in purgatory.

However, Hamlet does not extend that dread of murder to people lower down on the social scale. He has no scruples and no regrets about writing the letter that condemns the courtiers Rosencrantz and Guildenstern to death. He has found and opened a secret letter they are carrying that tells the English to kill Hamlet as soon as he lands on shore. However, they are simply the messengers: they have no idea what the letter, written by Claudius, says. That Hamlet would so blithely throw away the "little people" should give us pause; surely, he could have written a letter asking that they be imprisoned rather than killed.

Shakespeare, himself a commoner, seems to be making a comment on the blind spots of princes: they will look out for their own far more carefully than they will the common person. In Hamlet's case, as he is heir to the throne, this is all the more reprehensible: a good leader, like Oedipus, looks after the needs of the people.