Why does Brutus think that Caesar will become a tyrant in the future?
Very interesting question as it makes us consider Brutus' motives at this stage of the play for wanting to be involved in the plot against Caesar. 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.