One could argue that Caesar is the tragic hero. He possesses many of the characteristics of a tragic hero. He is a noble figure who experiences a downfall--a respected military leader who never lost a battle is outsmarted and assassinated by politicians. He possesses a tragic flaw (at least according to Brutus)--ambition. He experiences a tragic realization--before dying, he recognizes that Brutus, whom he thought of positively, has taken part in the conspiracy.
However, most critics identify Brutus as the tragic hero. Shakespeare presents Brutus in a positive light. He is noble; he comes from a family that helped establish Rome's republic. He is well respected (Cassius knows that he needs Brutus's status to convince other conspirators and the people or Rome that Caesar needs to be killed.). He experiences a downfall from his noble position. After the assassination in Act 3, Antony outwits Brutus in the funeral speech "competition," and Brutus is run out of town. He eventually has nothing--his life is in danger; his wife commits suicide; his house has been burned to the ground, and he has lost trust in Cassius, his closest ally from the conspiracy. Brutus also possesses a tragic flaw--poor judgment. He trusts the wrong people and then doubts them. He allows Antony to speak at the funeral and sway the crowd against the murderers. Before he knows for sure that the battle is over in Act 5, he chooses to die. His tragic realization occurs when he sees Caesar's ghost and recognizes that his killing Caesar has not bettered Rome.
In general, Brutus is more tragic because he loses everything. Caesar, on the other hand, gains much from being assassinated. His appointed heir eventually takes the title of Emperor. His people revere him even more after they hear the stipulations in his will. Finally, in reality, Caesar's battle strategies are still studied in military academies, and his name has inspired several words around the world such as "czar." His death ushered in the totalitarian rule that the conspirators were trying to prevent.