Basically I think "almost" everybody had their own justice to serve when it comes to Sephiroth. Cloud, well you know. Barret, his whole purpose is to save world and the souls who's seeking their way back to earth. Tifa basically has the same reasons as Barret. Cid, Nanaki and Yuffie basically has the same reasons as the two previous ones.
Vincent however has his own score to settle with Sephiroth, although Sephiroth didn't know that Lucrecia was his mother...and Hojo his farther. Vincent seeks justice...targeting Sephiroth when he actually should put his fully rage on Hojo instead. Cait Sith, wants to prove that his intentions are good...and that he actually cares about others.
Aeris' adoptive mother, has the justice of her deceased daughter Aeris...who Sephiroth kill. So basically everyone has a ligit reason for justice...I mean HE is a tread to the whole world !!!