In Julius Caesar, how does Brutus begin to justify joining the conspiracy against Caesar?
You have asked a question that goes to the heart of one of the themes of the play - the power of rhetoric and manipulation. The speech you want to examine very closely is Brutus' soliloquy in Act II scene 1, where Brutus tries to persuade himself into being involved in the conspiracy. If you were directing this play yourself, you need to think about whether Brutus is actually getting involved for the noble motives he possesses, or whether this speech is actually ironic: in talking about the danger of ambition and how this will probably lead Caesar to become despotic, Brutus is blind to his own ambition and how the same dangers await him.
Brutus starts off my saying there is no personal reason for him to kill Caesar:
It must be by his death; and for my part,
I know no personal cause to spurn at him,
But for the general.
The general good can be the only motive for assassinating Caesar, Brutus assures himself, before moving on to the crux of the issue:
He would be crowned.
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.
Note here how Brutus uses a metaphor of an adder to compare what might happen to Caesar if he is crowned. Crowning Caesar would give him more power, giving him a "sting" that he could do serious damage with to democracy.
Although Brutus acknowledges that Caesar has shown himself to be worthy of the power he has received up to now, Brutus falls back on aphorisms and "common proof" to persuade him of the danger were Caesar's power to grow bigger:
But 'tis a common proof
That lowliness is young ambition's ladder,
Whereto the climber upward turns his face;
But when he once attains the upmost round,
He then unto the ladder turns his back,
Looks in teh clouds, scorning the base degrees
By which he did ascend. So Caesar may;
Brutus thus uses such "common proof" to persuade him of the danger, though the use of words such as "may" of course perhaps makes us think that Brutus' conclusion is by no means certain, thus suggesting that other motives come into play that perhaps Brutus is blind to, such as his own ambition and envy of Caesar. Either way, by the end of the speech, Brutus has convinced himself:
And therefore think him as a serpent's egg
Which hatched, would as his kind grow mischievous,
And kill him in the shell.
Brutus and his conspirators must "kill" Caesar now before he "hatches" into the serpent that, according to Brutus, he will obviously become due to the amount of power he is gaining. His choice is made and Brutus has chosen his destiny.
Accessteacher is correct, it is in Act II, scene i that Brutus makes his final decision to join the conspirators. However, it is in his meeting with Cassius in Act I, scene ii that he begins to justify the action of murdering Caesar.
He hints at his own worries over the potential crowning of Caesar around line 40:
...Vexed I am
Of late with passions of some difference...
Which give some soil perhaps to my behaviours.
And then, when the crowd cheers offstage:
What means this shouting? I do fear the people
Choose Caesar for their king.
And it is the becoming King in the Roman Republic that Brutus will not tolerate. His justification for killing Caesar is already forming. He says to Cassius:
Set honor in one eye and death i' the other
And I will look on both indifferently.
For let the gods so speed me as I love
The name of honor more than I fear death.
So, he is willing to commit treason for the honor of holding up the true government of the Republic against tyranny, against a king.
And so he leaves Cassius, who now holds the firm hope that Brutus will join him in murdering Caesar for the good of Rome. When we next see Brutus in Act II, he has made his decision, the decision that he has alluded to in Act I.