By Aristotle's definition in his Poetics, the tragic hero is a man of noble stature whose fall is a result of his hamartia, which is NOT a tragic flaw. Rather, this is a criminal act committed in ignorance of some fact, or even for the better good of others. (e.g. Oedipus Rex kills his father not knowing that the man is his parent.)This act of hamartia differs greatly from the term "tragic flaw," a fault of character such as inordinate ambition, quickness to anger, jealousy, or arrogance.
Macbeth, then, would compare with other Shakespearean tragic figures as having a "tragic flaw," his overriding ambition, "vaulting ambition," as he himself terms it. Because he is so desirous of being King, he kills Duncan so he can be King. However, Macbeth does not compare with Oedipus Rex, for he commits no act of harmartia.
A tragic hero begins as a hero but has a tragic flaw. Macbeth's flaw was his pride and arrogance. He believed he deserved to be king, even though there was no reason for him to take the throne. The witches told him he would be, and he wanted it. Thus he destroyed himself, because he went from hero to murderer by the end of the first act.
One of my Classical Lit students just wrote an outstanding paper about Seneca's Medea and Oedipus Rex by Sophocles. These would be excellent examples for you to research for the portion of your assignment pertaining to ancient Greece. In addition, they both compare/contrast nicely with Shakespearean tragic heroes like Macbeth and Othello. Be sure to check the eNotes information on all of these plays - that will help you a great deal in your research! Good luck!
Be sure to remember that tragic heroes all have what's known as a tragic flaw in their characters, which brings about their downfall. This will be a common characteristic of a tragic hero no matter the cultural context.