Are you sure you mean Antony? He is not generally examined as a tragic hero in the play. Brutus is clearly a tragic hero, and some have made arguments that Caesar is a tragic hero, but Antony does not seem to fulfill the role as Shakespeare developed it.
The Shakespearean tragic hero is a powerful figure, even admirable, who occupies a high place in society before he falls into destruction. His tragedy is that he is destroyed by a fatal flaw within his own character, a flaw that he does not even recognize until it is too late. Macbeth, Hamlet, Othello, King Lear, and Brutus certainly fit this characterization.
Antony, in contrast, does not fall from a high position, nor is he destroyed in Julius Caesar. In fact, his character experiences the opposite fate. In the beginning of the play, he is shown to be a shallow, inconsequential young man who devotes himself to games and revelry. After Caesar's murder, however, he becomes focused on revenging Caesar's death and then grabbing power for himself. By the end of the play, he has formed a powerful military alliance, waged war, and defeated the army of Brutus and Cassius. Antony is the victor in the power struggle that resulted from Caesar's assassination. Brutus, however, who tried act in the best interests of the Roman people, is destroyed by his own naive nature and disastrous decisions.