The clearest example of ambition in the play can be found in Caesar. Julius Caesar is assassinated, in large part, because of what the conspirators saw as his great ambition. He wanted to become the supreme ruler of Rome and to end its democracy, while "...the conspirators are champions of popular rule."
The conspiracy aimed to end this ambition and to ensure that the political body of Rome remained democratic in principle.
Immediately after Caesar is slain, Brutus proclaims to his fellow conspirators that "ambition's debt is paid" (III.i.82).
Caesar's goal is the first and most clear example of ambition in the play, but not the only act that can be read as a symptom of ambition.
The conspiracy seeks to ensure that the republic remains democratic in nature, however, those who lead the conspiracy stand to gain considerable power for themselves after Caesar is killed. Brutus is talked into participating via a play upon his principles and his vanity.
Brutus feels that he is capable of leadership and he expects to gain some power for himself when Caesar is removed.
Finally, Marc Antony acts on his own ambitions, grasping the mantle of leadership and using the conspiracy's actions as a convenient launch for his own vault into leadership.
...the most ambitious of the play's characters is not Caesar or Brutus, but Mark Antony, who exploits the situation at hand...
The play Julius Caesar is, to a large degree, about ethics and ambition in politics. Rome had, for many centuries, been a Republic but that form of political organization masked a problem that in fact much of the control of the state was in the hands of a patrician oligarchy dominating the Senate. Caesar, as many of the subsequent emperors, played off the plebeians against the patricians using "bread and circuses" to bribe them to essentially cede the little voice they had in the government.
The first character who is ambitious is Caesar himself. He is a somewhat ambiguous figure though, as he is actually a decent leader, but his becoming what was essentially a dictator marked the end of a participatory political process, although in Caesar's favor it can be said that the Republic had become essentially dysfunctional when he formed the first triumvirate. The degree to which Caesar was genuinely a champion of the plebeians vs. acting out of personal ambition is debatable.
Of the conspirators, Cassius especially and most of the others, are members of the patrician faction whose opposition to Caesar has to do with personal ambition, greed, and jealousy, as Brutus points out in the lines:
Let me tell you, Cassius, you yourself
Are much condemned to have an itching palm,
To sell and mart your offices for gold
The one exception among the conspirators is Brutus, "the noblest Roman of them all", who acts out of a sense of ethics and a belief in the ideal of the Republic.