How is Hamlet a tragic hero?

A tragic hero is more than just the hero of a tragedy who dies in the end. When we identify a tragic hero, we look at a character who loses everything and everyone close to them over the course of the play, usually because of one of their own flaws, or hubris.
Hamlet has many strong character qualities. At the beginning of the play, he's loyal to his father and vows revenge for his death. He feigns insanity to convince the people around him that he is crazy while he investigates the ghost's claims that Claudius is the killer.
However, Hamlet's once-noble intentions fall apart as his desire for revenge consumes him and as each event is linked to his inevitable death. We see him mistakenly kill Polonius and simply shake off the murder:


Thou wretched, rash, intruding fool, farewell.
I took thee for thy better. Take thy fortune.
Thou find’st to be too busy is some danger.



His cold and harsh language towards Ophelia could be the cause of her madness and even suicide; he convinces her that he never loved her and warns her not to become a "breeder of sinners" (act III, scene 1). Once he realizes that she is dead, he sees what he has done, but it is too late to stop, and he continues his downward spiral by accepting a duel with Laertes. This duel has nothing to do with his revenge, but he is tricked into accepting the duel: Osric details just how good Laertes is at fencing, but he says that he believes Hamlet is better.

The duel serves as Hamlet's final action. Before he is killed by Laertes's poisoned sword, he kills Laertes with the same sword. His mother dies by accidentally drinking poison—the poison Claudius intended for Hamlet—and Hamlet both stabs his stepfather and forces him to drink poison.

As a tragedy, we know from the beginning of the play that things will not end well for our main character, but as a tragic hero, we can link his decisions and events to clearly lead to his inevitable death.