In the play Julius Caesar, how did Cassius persuade Brutus to join the conspiracy?
Cassius attempts to convince Brutus to join the conspriacy in a couple of ways, though the more effective way is through deception.
First, during the feast of Luprical in the first act, Claudius points out that Caesar is no more worthy of being crowned emporer than Brutus. He points out Caesars many physical weaknesses, such as his girlishness when he is sick or his inability to swim for distances. These weaknesses, however, are not enough to convince Brutus.
Casius knows that Brutus' first concern is for the people of Rome, so to convince him to kill Caesar, Casuis must first convince him that Caesar is bad for Rome. He plants the idea that Caesar is too powerful and that power causes people to be corrupt. He then convinces Brutus that the people themselves fear Caesar and his control.
To do this, Casius forges several letters from anonymous Roman citizens and has them placed around Brutus' home where he is sure to find them. These faked letters convince Brutus that the people would be better served if the threat of an all-powerful Caesar were removed.
For more information about the characters and themes in Julius Caesar, see the links below. I've also included a link to the soliloquy where Brutus decides that he must stop Caesar and why.
He would be crown'd:
How that might change his nature, there's the question.
It is the bright day that brings forth the adder
And that craves wary walking. Crown him? that;
And then, I grant, we put a sting in him
That at his will he may do danger with.
Brutus is saying that though Caesar may be a good man now, too much power may change his nature, and turn him into something dangerous.