Limit breaks are pretty tough to figure out at first, and they may seem very mysterious and elusive to the beginning player. However, there is a set system to getting these. It's done like this: Every character begins with one limit break, and the next one is acquired by using the first one 8 times (this doesn't have to be during one battle or within a specific length of time--the character can even leave the active party, and the amount of breaks he/she has used already will not change). Then, to get the Level 2 Limit Break, the character must defeat (actually deliever the killing blow) 80 enemies. The process repeats for every level.
Let's use Cloud as an example. He begins the game with Braver. After Braver is used 8 times, he gains Cross-Slash. After that, he must kill 80 enemies to gain Blade Beam. For Climhazzard, Cloud must use Blade Beam 8 times, and for Meteorain, he must then kill another 80 enemies. I hope that makes sense to everyone.
The only exceptions to this rule are Cait Sith and Vincent. After killing 80 enemies, Cait Sith receives his only other Limit Break--Slots. As for Vincent, he only has one Limit Break per level, and each one is gained after he has killed 60 enemies