To level up a limit break, your character has to use their respective level limit break a certain amount of times (these numbers are approximate and vary slightly with character).
Level 1/1 must be used 10 times.
Level 2/1 must be used 8 times.
Level 3/1 must be used 6 times.
Level 4 must be obtained at a world map location and used on the character.
The level x/2 limit must kill 40-80 enemies (based on the character and general limit break level) to acquire an entirely new limit break level.