I think that's actually a very challenging question. As I see it, there's an easy answer, and a difficult answer.
The easy answer is that the conflict is whether or not the conspirators are going to successfully manage to kill Caesar. There are a whole series of moments where Shakespeare makes you wonder whether they will get away with it, from the first moment Cassius persuades Brutus that Caesar might pose something of a threat to Rome and should somehow be dealt with, through Brutus' agonisings in his orchard, through Portia's worrying before the scene of Caesar's death (worrying that the conspiracy might be discovered), right up to this moment in the murder scene:
I wish your enterprise today may thrive.
What enterprise, Popilius?
Fare you well.
What said Popilius Lena?
He wish'd today our enterprise might thrive.
I fear our purpose is discovered.
This conflict is obviously resolved in Act 3, Scene 1, when Caesar dies.
Yet I'd argue the conflict is more complex. The real question is - was it right for them to kill Caesar? And that question resounds throughout the play. Brutus changes his mind - initially he thinks it's a good idea, but he tells Cassius that he should "be sorry" for the deed. And Shakespeare never resolves it: at the end of the play there is no hint as to whether Brutus and Cassius have done Rome a service, or committed a horrible crime. It's an unresolved conflict of right against wrong.
In "Julius Caesar," the main conflict resides within the tragic character of Brutus, whose idealism brings about the death of the man whom he loves, but loves less than he loves Rome. For Brutus perceives Caesar as a tyrant who will harm Rome. The irony is that by killing Caesar in his love for Rome, Brutus unleashes a brutal civil war as well as many unhonorable actions. When Brutus argues with Cassius about taking bribes in Act IV, he condemns Cassius's behavior:
I had rather be a dog, and bay the moon,/Than such a Roman (IV, iii, 26-28).
Brutus goes on to tell Cassius that he is
armed so strong in honesty/That they [Cassius's threats] pass by me as the idle wind...When Marcus Brutus grows so covetous /To lock such rascal counters from his friends/ be redy, god, with all your thunderbolts,/Dash him to pieces
Finally,when Brutus realizes that the civil war and the resulting triumvirate of Marc Antony, Octavius Caesar, and Lepidius is far worse, he runs onto his sword and dies, "dashing him[self] to pieces. A tragic end, indeed.
Honestley i believe that the true conflict of this play was that ceaser was going to use the people of rome as slaves.