Yes and no. There are varying definitions of the "hero" in literature, the most basic being fairly synonymous with protagonist, in which case, yes, Hamlet is a hero. A popular interpretation of him, however, is that of the anti-hero due to the fact that, although he is the protagonist, and also arguably a tragic hero, he also has certain negative sides to his character, such as violence, vulgar language, etc.
Exactly! A normal man, and that's what I think is the marrow of the play. I think Shakespeare foresaw the "fall of the grandiose man" of the Classical period. Don't you think that Hamlet totally lacks that freewill that determination so typical of classical heroes of the western culture. Remember Ajax's determination to silence his shame with suicide or Antigone's pride and will to honour the dead. Hamlet is coerced into action. A simple sign was enough for Orestes to rush to Argos and end the whole affaire. Hamlet seems more like a character of a modern piece of drama. There is an utter absence of determination. His revenge against Claudius was triggered by another's vengance. I have always admired the character of Hamlet but more like a stoic philosopher more than a classical hero.
I think 'heroes' are rather unlikely, melodramatic characters. Big Muscles, Decisive Action, Unshakeable Moral Clarity etc etc. Heroes are simplistic devices with which an author streamlines reality and creates a 'super-man' to show us the ideals that we should aim for.
Shakespeare places Hamlet in a typically 'heroic' situation, the basic plot sounds like a movie trailer...
"The good son... of a murdered king... in a world turned upside-down. A quest sent from The Grave... A mother's betrayal... Only one man knows The Truth... Only one man can revenge his noble father... Only Hamlet can fight for his freedom. To be, or not to be... A Hero. Certificate 15, Coming Soon."
But the point is that Shakespeare wouldn't cast Vin Diesel as Hamlet. If Hamlet were a 'classical' hero, He would act. He would know what to do. He wouldn't greive. He wouldn't hesitate. He would march up to Claudius, happily stick a knife in his throat then get an army, successfully invade Norway and kill Fortinbras. Job done, Hero Status confirmed. The End.
But if Shakespeare had written that. The play would be lying in the bottom draw of some University library, a minor curiosity and nothing special.
Hamlet is a sensitive normal man placed in a super-hero situation. He is 'The Reluctant Hero'.