To go from one Level 1 Limit Break to another Level 1 Limit Break, you need to use that first limit EIGHT times.

To go from one Level 2 Limit Break to another Level 2 Limit Break, you need to use that first limit SEVEN times.

Ditto that for Level 3 except it's SIX times.

To go from Level 1 to Level 2, and from Level 2 to 3, you just need to keep dealing the killing blow to enemies.