How does Brutus change his mind about the need for Caesar's death?
I'm not sure that Brutus necessarily "changes his mind" about the need for Caesar's death. Instead, it seems more appropriate to discuss ways in which Brutus is convinced that Caesar's death is necessary, as there is no point in the play at which Brutus vows not to kill Caesar and then changes his mind.
As the play opens, audiences observe a troubled Brutus, who tells Cassius that he has been "with himself at war." He doesn't, though, not provide Cassius with a more specific answer. Cassius, shrewdly guessing that Brutus's upset has something to do with Caesar's growing power, begins laying the groundwork to recruit Brutus to join the conspiracy.
Cassius is well aware of Brutus's love for Rome and his desire to do what is best for the "general good," but is also aware of Brutus's respect for and love of Caesar. To convince Brutus that Caesar is becoming too powerful and must be murdered, Cassius flatters Brutus, villifies Caesar, and even forges notes to Brutus that appear to have come from concerned citizens expressing fear about Caesar's growing power.
By Act 2, scene 1, Brutus has already decided to join the conspiracy to murder Julius Caesar. Despite his love for Caesar, Brutus is convinced (and relatively easily) that he must murder Caesar for the good of Rome.