In Julius Casear, how does Cassius finally convince Brutus that Caesar should be killed?
Firstly, and mainly, he does it by persuasion. Have a look at Act 1, Scene 2, and look at the speeches Cassius makes to Brutus: he makes arguments against Caesar, based on Caesar's ambition, the fact that Rome is supposed to be more powerful than any individual, based on Caesar's own girlishness and weakness.
Moreover, Cassius is excellent at playing on Brutus' own sense of arrogance and self-belief. Look at thsi extract from Act 1, Scene 2:
The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings.
Brutus, and Caesar: what should be in that Caesar?
Why should that name be sounded more than yours?
Write them together, yours is as fair a name;
Sound them, it doth become the mouth as well;
Weigh them, it is as heavy; conjure with 'em,
Brutus will start a spirit as soon as Caesar.
It poses as an abstract example, but of course it isn't. It tries to say "Why is Caesar's name so much more powerful than anyone else's?', but subtly it is saying 'Brutus, why is Caesar's name worth more than YOURS'. Brutus' ambition and arrogance is played on: and, that, I am quite sure, is a key factor in why he actually gives in.
Cassius also isn't above resorting to underhand tactics:
Good Cinna, take this paper,
And look you lay it in the praetor's chair,
Where Brutus may but find it; and throw this
In at his window; set this up with wax
Upon old Brutus' statue.
We see Brutus receive one of these letters in Act 2, Scene 1, and it certaintly works on him. 'Brutus, thou sleep'st, awake', it says, and Brutus seems to agree.
Hope it helps!