Why does Cassius want Brutus to kill Caesar?

Cassius is a very clever character. He knows that if he and his associates conspire against the popular and beloved Caesar, the Roman people will respond with rage and violence. However, Cassius also knows that if he can convince the well-respected Brutus to join him, the Roman people will assume his plans are righteous. As his coconspirator Casca says,

O, he [Brutus] sits high in all the people’s hearts,And that which would appear offense in usHis countenance, like richest alchemy,Will change to virtue and to worthiness. [Act 1, Scene 3, lines 162-5]

Cassius replies that "Him and his worth and our great need of him / You have right well conceited" (166-167).
 
Cassius's general character type should be familiar to readers of Shakespeare: he has the same sort of subtle cleverness and love of deceit as other villains like Iago and Richard III. As Caesar himself says of Cassius:

Such men as he be never at heart's easeWhiles they behold a greater than themselves,And therefore are they very dangerous. [Act 1, Scene 2, lines 218-20]

Unfortunately, the noble and trusting Brutus is deceived by Cassius, who manipulates him from the very start. Cassius subtly shames Brutus for appearing distant and preoccupied (in classical Rome an important part of manly virtue was being fully engaged in public). He also amplifies Brutus's great fear: that "the people / Choose Caesar for their king" (Act 1, Scene 2, lines 85-6).
Later, Cassius plants forged letters that complain of Caesar's ambition and desire to become a tyrant in Rome. By the time Cassius and the other conspirators arrive at his house, Brutus is ready to join them. However, he insists that Mark Antony's life be spared.
Of course, the decision to spare Antony becomes the conspirators' undoing, as Antony is able to win the people to his side in his famous oration. Antony also teams with Octavian to defeat Brutus and Cassius on the battlefield. However, Antony is wise enough to say of Brutus:

This was the noblest Roman of them all.All the conspirators save only heDid that they did in envy of great Caesar (Act 5, Scene 5, lines 74-6)
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