A tragic hero cannot overcome the forces that oppose him and his death affects the course of an empire.
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I'm going with Brutus here. He's got the perfect resume for a tragic hero. He's a decent guy but really does not understand his flaws and limitations. He ends up destroying himself and those around him. Brutus loves Caesar but he loves Rome more. So, he ends up in a nasty conspiracy. Brutus's flaw isn't that he is an honourable guy rather than he thinks others are honourable as well. He cannot see the daggers beneath men's smiles and is easily manipulated by Cassius, “Three parts of him Is ours already, and the man entire upon the next encounter yields him ours." Soon after the funeral Antony turns the crowed against the conspirators and Brutus is pretty much done.
What happened to Brutus was tragic. However, if you look at the play only in the sense of the genre tragedy, then you have to see Julius Caesar as the tragic hero.