Julius Caesar refused the crown because he did not want the people to think he was a king.
During the Feast of Lupercal, Brutus and Cassius are told by Casca that the cheering they overheard was caused by Caesar pandering to the people. Casca sneeringly describes the scene, how Mark Antony, Caesar’s deputy, offered Caesar a crown, and Caesar refused it, and "the people fell a-shouting."
Brutus accuses Caesar of ambition, but according to Casca, he refused the crown that Mark Antony offered him a total of three times!
If Caesar wanted to be king so badly, why refuse a crown? Actually, Caesar did not want to be a king. He had the title of dictator, which he is never called by Brutus and his minions (or Shakespeare) and actually got very angry when the people called him king. Being a dictator sounds worse than being a king, but it was a legal title given by the senate. It was usually temporary, and it was intended to help him ensure the rule of law.
Cassius uses this incident to get Brutus on his side.
When you are looking for people to get on your side for a conspiracy, it can be a bit tricky. How do you know who to trust? This is especially true when the person you are feeling out is an aristocrat known to be close to the person you are trying to betray.
Cassius had to be very careful before trying to turn Brutus. He had to make sure Brutus really was questioning Caesar’s ambition. Brutus confirms that he is worried that Caesar wants too much power. Cassius knows then that he can safely talk to him about joining the conspiracy.
I suppose the question remains: Why did Antony offer Caesar the crown? Why did he offer it to Caesar three times? Part of this knowledge is lost to history, unfortunately. Historians have argued that Antony might have been put up to this by the senate, which was trying to make Caesar look bad. The number is easier to answer. It was playacting. Antony loved to play to the crowd.
Antony may seem like a loyal supporter of Caesar, but may have thought he was doing a good thing. Who wouldn't want a crown? Or, he may have been acting on his own ambition. After all, he was definitely an ambitious man himself. Some people have even speculated that he was part of the plot to kill Caesar, either because he found out he wasn't Caesar's heir, or because he wasn't rising fast enough.
Some people said that Caesar refused the crown because it was too small. Others said he was waiting. Decius Brutus offers Caesar a crown again, and Caesar does not refuse it.
Why refuse the crown so publicly, but not refuse the senate's? Although Caesar does not answer, he does not say he would accept the senate's crown either. He might have refused that one too. It also might be a different kind of crown. The Romans had many crowns. There were grass crowns, civic crowns, and laurel wreaths. I am sure Caesar had plenty of those and would have accepted others. He just does not want to offend the senate by ignoring them.
Whatever the reason for Caesar not accepting the crown, this incident was one of the nails in Caesar's coffin. It gave Caesar's opponents more proof that he was tyrannical or had ambition. As Shakespeare shows, it could be a recruiting tool for the conspiracy. Whether Antony meant to betray Caesar intentionally or not, he inadvertently did by offering that crown.