How is Antony a tragic hero in Shakespeare's Julius Caesar?
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.
You may get a differing opinion after this answer, but I would suggest Antony is not a tragic figure in Shakespeare's Julius Caesar. A tragic figure possesses a tragic flaw that leads to his eventual downfall. Antony wins in this play. Brutus is the tragic figure. Brutus misjudges situations and people and makes decisions that lead to his failure. Brutus is noble and loyal to Rome. He is the conspirator that actually assassinates Caesar for "pure" reasons, for the good of Rome. But when he allows Antony to speak at Caesar's funeral his bad judgment allows Antony to sway the Roman mob in Antony's favor and eventually leads to Antony's victory over Brutus in the civil war. The play centers much more on Brutus than it does on Antony. Brutus is the tragic hero.
I don't believe Antony to be a tragic figure in Julius Caesar because he doesn't have the rise to power (Antony already has his power), his fatal flaw (as far as I know he doesn't have one that leads to his downfall), or his death (he doesn't die in Julius Caesar).