By definition, a tragic hero has a tragic flaw--the theory that there is something inherently wrong with that amazing person which causes his or her downfall. By this definition, I concur that Julius Caesar could be considered a tragic hero.
Did he undoubtedly accomplish amazing things in his life and the play? Yes. Did the people love him? Yes. They wanted him for their emperor, and offered him the crown in the play three times.
However, according to Cassius and the other conspirators who finally convinced Brutus, Julius Caesar was ambitious to a fault. They believed that Caesar wanted the crown and would misuse its power to serve himself rather than serve the public as he was sworn to do in his service as a senator. It is without a doubt that the other conspirators were also jealous of Caesar's popularity and accolades, but Brutus agreed to be part of the murderers only for the protection of his beloved Rome.
Therefore, Julius Caesar dies for his ambition (a tragic flaw) and Brutus dies because he is gullible and easily led into the brotherhood of the conspirators without full proof of Caesar's intentions.