Historically, Julius Caesar's robe would be his best toga, the one he would wear to a meeting of the Roman Senate. It would look like the toga depicted in the statue of Caesar Augustus, below.
However, it may be that no one in Shakespeare's time had an accurate idea of what a roman toga looked like. See, for example, sketch below, known as "the Peacham drawing." It dates from about 1595 and depicts characters in another Shakespearean play set in Rome, Titus Andronicus. Titus, on the left, holding a staff, is a Roman general. Behind him are figures probably meant to represent two of his soldiers. Titus is wearing clothes that look vaguely Roman, but the soldiers, in their puffy breeches, are purely Elizabethan. No one in Rome ever wore any such thing.
Therefore, what Shakespeare may have had in mind was the kind of robe that was worn by ministers of state in Shakespeare's time, such as the one depicted below in the portrait of William Cecil, Lord Burghley, the chief advisor to Queen Elizabeth I.