Aristotle's definition of tragedy is best seen in the quote:
Tragedy, then, is an imitation of an action that is serious and complete, and which has some greatness about it. It imitates in words with pleasant accompaniments, each type belonging separately to the different parts of the work. It imitates people performing actions and does not rely on narration. It achieves, through pity and fear, the catharsis of these sorts of feelings. (1449b21–29)
One thing he means here is that, as a work of literature enacted, a tragedy imitates what could be real-life actions. The historical account of Julius Caesar's assassination could not be considered a tragedy, even though it might have been tragic, because it is an account of the actual actions rather than being an imitation of the actions. However, if a group of performers enacted his death, then that would be a tragedy. This also helps explain his distinction between "narration" and "drama." A narration is an account of events, like an historical account. Drama, on the other hand, takes those historical events and interprets them from a universal perspective. Drama shows us a "clear cause-and-effect chain," showing us how the historical, tragic events are universally relevant, relevant for all mankind in all generations (McManus). Shakespeare's Julius Caesar is an excellent example of Aristotle's definition of tragedy because it takes the historic event of Caesar's death and draws from it the question whether or not killing a king can be justified if it may put an end to tyranny, thereby showing us the cause-and-effect relationship of Caesar's assassination. In addition, since a tragedy must be a drama, Aristotle defines that it must also have a plot. A plot must have a beginning, middle, and a conclusion.
Finally, Aristotle argues that a tragedy must also "achieve catharsis" through the feelings of "pity and fear." Catharsis is the relief of emotions or "emotional tension" (Random House Dictionary). His argument is that, since we are drawn into feeling the emotions of pity for the characters, just like we feel pity for Caesar, Brutus, and Antony in Julius Caesar, then we release our bottled up feelings of pity. Likewise, since we feel the characters' fears, we also release our own emotions of fear.
Hence we see that in multiple ways, Shakespeare's Julius Caesar, as well as many other tragedies, perfectly fit Aristotle's definition of tragedy.