What is the main conflict in Julius Caesar and how is it resolved?

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:

POPILIUS:
I wish your enterprise today may thrive.

CASSIUS:
What enterprise, Popilius?

POPILIUS:
Fare you well.

BRUTUS:
What said Popilius Lena?

CASSIUS:
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.