The previous answers both link to the theme of 'public good versus personal benefit'.  Brutus kills a friend for the sake of Rome.  He puts his country's needs before his own.

Another theme that  is strongly demonstrated through Brutus is  'lack of self-knowledge'.  Brutus is manipulated into believing that Caesar will become a tyrant and destroy Rome as a republic. (Remember that the term republic is not what we define as a 'republic' in modern era - the Roman republic did not give commoners voting rights for instance.)

Brutus allows Cassius to be 'his mirror' and to dictate his thoughts and emotions.  Cassius cleverly flatters Brutus by telling him that Caesar is no better than Brutus, Caesar is not a god, but Brutus has the admiration and support of the people of Rome.  The Romans are looking at Brutus to save them from Caesar' tyranny.

If Brutus was more in touch with his own strengths and weaknesses, he would have identified Cassius as a manipulator.  He would have been able to question why Cassius hated Caesar so vehemently that he would attempt to humiliate Caesar to the extent where he compares him to 'a girl'.  Cassius launches a personal attack on Caesar.  Brutus should have been able to question this since he was a man with 'grey hair'  (wise) at this stage.  Yet, he does not, he accepts everything Cassius tells him and joins the conspiracy despite the fact that it troubles him so much that he is unable to sleep or find peace of mind.

This reveals that Brutus does not understand enough about his own weakness, he is unaware of the fact that he is easily manipulated. He does not realise that he is being used by Cassius to reach a private goal (destroying Caesar out of envy) rather than acting for the good of Rome.  Shakespeare confirms this notion when Rome is destroyed after the killing of Caesar.  Both Cassius and Brutus die as well which leaves the audience with the question as to whether any of this would have taken place if Brutus did not allow himself to be manipulated by a man (Cassius) who was not thinking about the good of Rome in the first place.