The play does not answer: through Casca's voice it tells that Caesar refused a crown, and it shows others believing he wanted it, but it never shows him accepting it or declaring that he wants to be king.
The crown is offered at the Lupercal (Act I, scene 2), but offstage. Casca tells Brutus and Cassius that Antony offered it three times and that Caesar pushed it away three times, each time more gently, with the people shouting at each refusal. Casca says he did not pay attention, since it was all nonsense, but thinks Caesar wanted it. He also tells that Caesar fell in the square and foamed at the mouth.
In the stormy night (Act I, scene 3), Casca says the senators intend to set Caesar up as king the next day, anywhere except in Italy. On the morning of the Ides (Act II, scene 2), Decius uses the offer to persuade him to leave home: the Senate had decided to give him a crown that day, and Caesar yields.
After his death, the subject returns. In the Forum (Act III, scene 2), Antony reminds the people that Caesar refused the crown three times and asks whether this was ambition, and a citizen concludes that, if he refused it, he was not ambitious. Brutus, however, had already decided in Act II, scene 1, that Caesar must die for what the crown would make him become.