Julius Caesar is very obviously seeking to become a king or emperor and to have despotic rule over Rome and the empire. He has an army behind him. Numerous people fear and hate him. In the opening scene two tribunes are driving commoners off the streets and tearing down decorations in an attempt to undermine Caesar's growing popularity. This establishes the conflict.
Shakespeare is following actual Roman history and cannot impose his own imagination too heavily on the conflict or its outcome. When Cassius leads a successful coup, that would seem to be the end of that conflict. However, Antony becomes the dead Caesar's alter-ego and leads a counter-attack against Cassius, Brutus and the other conspirators. Caesar's ghost appears to Brutus twice, evidently to impress the audience that Caesar is still an awesome and mighty figure even after his death. When Brutus and Cassius are defeated at Philippi by Antony and Octavius, both acting as Caesar's representatives and avengers, Brutus says
O Julius Caesar, thou are mighty yet.
Thy spirit walks abroad, and turns our swords
In our own proper entrails.
So the conflict throughout is between Julius Caesar and those who oppose his powerful will to establish an imperial dynasty.