A tragic hero is one, who is otherwise above reprove and perfect in character, but who due to one or some falliability or flaw in character brings about his own downfall or a turn of fortunes. Though the story of the Prince of Denmark does end in a tragic conclusion, with everybody dead, there has been some disagreement as to whether Hamlet can be called the archetypal tragic hero?
One school of thought would give it that he is not. As his character has been shown to be nearly perfect. He loves his father and is shocked at the hint of the treacherous murder, by the brother, his uncle. He is even more hurt at the fearful prospect of doubting his mother to be also a party to this crime and of committing incest. And he does not immediately believe the anti-christian thought of his father's ghost roaming the kingdom. He shows love, loyalty and sensibility too and revenge is not his option, but he wants his uncle and mother to commit their guilt.So his tragic end is only circumstantial.
The other school would say that the indecision that Hamlet shows in taking his revenge is his irrevocable flaw and the cause of his tragic demise. They would not lessen his fault by calling it a temporary insanity or shock but say it is procrastination and part of his nature, and hence tragic in consequence. In all his soliloques we see the confusion of his mind, and we realize he is not able to make sense of the world around him and so no amount of motivating incidents or causes is able to clear his hesitation and he is as per his words 'most dreadfully attended' - the tragic hero.