I'm always surprised by how many people don't know this.

1-1: Each character starts with this one.
1-2: The character must have performed his 1-1 Limit Break eight times.

2-1: The character must have killed 80 enemies.
2-2: The character must have performed his 2-1 Limit Break eight times.

3-1: The character must have killed an additional 80 enemies.
3-2: The character must have performed his 3-1 Limit Break eight times.

4-1: You must acquire this Limit Break as an item, and the character must have learned all his previous Breaks (1-1 through 3-2) before you can apply it to him.

Exceptions: Cait Sith only has two Limit Breaks. He starts with Lucky Dice and has to kill 80 enemies to learn Slots. Vincent only has four Breaks. He starts with Garian Beast, must kill 60 enemies to learn Death Gigas, must kill another 60 enemies to learn Hell Masker, and his final Break is learned by acquiring it as an item and then applying it once he's learned the previous three.