How is the conflict between the public and private portrayed in Shakespeare's Julius Caesar?
What strikes me most about the concept of public vs. private conflict is Marc Antony's traitorous behavior—first against Brutus.
Shakespeare's Julius Caesar is a fascinating study into human nature and how ambition manifests itself in people. Cassius and the other conspirators want Caesar dead not because they want his position or power; their reasoning is more childish: they are jealous of Caesar and feel they have been slighted by him.
The minor conspirators of the plot are generally motivated by dissatisfaction with Caesar's high-handed treatment of them and by personal grievances.
Brutus, on the other hand, is fearful of what Caesar will become, for he is sure that Caesar will agree to be crowned "king" of Rome—and become a tyrant—and in doing so, Rome will suffer. Brutus loves Rome more than Caesar (who he truly does care for), and so he agrees to take part in the assassination for the sake of Rome.
Enter Marc Antony. If we are looking for a villain, it is in the character of Antony. He is, if nothing else, a man who knows how to capitalize on the chaos of the assassination so that he will eventually become one of the "ruling triumvirate"—three leaders of the Roman Republic (along with Octavius—Caesar's son).
After the assassination of Caesar, in which Brutus readily admits his part, Antony visits the assassins. Openly to them, he shakes each of their hands. Cassius asks whose side Antony is on, and Antony says:
...I took your hands...
Friends am I with you all and love you all... (III.i.234, 236)
Antony notes that he really only wants to speak at Caesar's funeral. Against Cassius' objections, Brutus agrees—he feels that since he (Brutus) will speak first, he can prepare the crowd for whatever Antony may say—letting the public know that Antony has their blessing/permission to speak.
…I will myself into the pulpit first,
And show the reason of our Caesar's death.
What Antony shall speak, I will protest
He speaks by leave and by permission... (255-258)
Brutus, Antony and the others all part company, as Antony remains behind. It is out of their hearing—revealing a private conflict—that Antony reveals his true plan:
Woe to the hand that shed this costly blood!
Over thy wounds now do I prophesy…
To beg the voice and utterance of my tongue,
A curse shall light upon the limbs of men… (278-279, 281-282)
At the funeral, after Brutus delivers his speech, Antony gets up seemingly to praise Caesar, but also to condone the actions of the conspirators. However, while he at first appears to support these men, he eventually convinces the crowd that Caesar has been murdered and the crowd turns on Brutus and the others. This is the start of civil war as Antony had predicted. Antony may have supported Caesar in life publically, but privately, he is interested in seizing power—which he accomplishes at the end of the war, with the deaths of Cassius and Brutus.
In a smaller sense, Antony also publically accepts Lepidus as part of the new ruling triumvirate, but insults him privately with Octavius:
You may do your will,
But [Lepidus is] a tried and valiant soldier.
So is my horse, Octavius, and for that
I do appoint him store of provender. [feed] (IV.i.30-33)
In public, Marc Antony acts in a positive manner. In private, his thoughts are very different, tied to conflicts in the play. With Brutus (during the play) the conflict between the two arises from what is public and what is kept private.