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.