Since Brutus has no personal grudge against casser of what must he convince himself before he can take part in casears assassination
In Shakespeare's Julius Caesar, Brutus is the conspirator that is also a noble Roman and has the good of Rome in mind. The other conspirators appear to be motivated by ambition or envy or greed--at least the ones we get information about.
Brutus, as you say, does not have a personal grudge against Caesar. What Brutus must convince himself of is that Caesar may become a tyrant and turn Rome, if he is crowned emperor, into a dictatorship instead of a republic.
Brutus weighs the possibilities and ultimately decides that the risk is not worth taking. He will not take the chance that Caesar will become emperor and then a tyrant. He joins the conspirators in assassinating Caesar.
Brutus may be wrong or he may be right, but given the terrible decisions he makes throughout the rest of the play, one might suspect he decides in error here, as well.