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, Caesars 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!
 
    CASSIUS
    They shouted thrice: what was the last cry for?
    CASCA
    Why, for that too.
    BRUTUS
    Was the crown offered him thrice?
    CASCA
    Ay, marry, was't, and he put it by thrice,
    every
time gentler than other, and at every
    putting-by
mine honest neighbours shouted. (Act 1, Scene
    2)
 
  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.
            BRUTUS
     What means this shouting? I do fear, the
    people
Choose Caesar for their king.
    CASSIUS
    Ay, do you fear it?
Then must I think
    you would not have it so. (Act 1, Scene 2)
 
 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
    Caesars 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.
 
    I have, when you have heard what I can
    say:
And know it now: the senate have
    concluded
To give this day a crown to mighty Caesar. (Act 2, Scene
    2)
 
 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.