This article is from the Final-Fantasy and other Square Soft Games FAQ, by firstname.lastname@example.org (Nick Zitzmann) with numerous contributions by others.
The answer is highly debatable, but what's listed here is what seems to
be the best method to learning new Limit Breaks:
Level 1, first Limit Break:
* You start with this one, so don't sweat it.
Level 1, second Limit Break:
* You must use your first Limit Break of that level ten times.
Level 2, first Limit Break:
* Kill a certain number of enemies. You don't have to win that battle,
you just have to kill a particular enemy using the character who's
trying to learn a new Break. You can kill them in any way, too;
use magic, use Morph, use an item, whatever. The number of enemies
you have to kill tends to be between 40 - 80. It's been proven that
Vincent can kill less enemies to learn his second Breaks, but the
exact number is unknown.
Level 2, second Limit Break:
* You must use your first Limit Break of that level eight times.
Level 3, first Limit Break:
* See 'Level 2, first Limit Break' notes.
Level 3, second Limit Break:
* You must use your first Limit Break of that level six times.
Level 4, Extreme Limit Break:
* You must know all your other Limit Breaks. Then, select your
character's 'Extreme Limit Break' manual and use it on them just
as you would any other item. See below for details on finding
your character's Extreme Limit Break manual.
Note that Limit Breaks work differently from character to character...
Cloud, Barret, Aeris, Yuffie, Red XIII, and Cid all have two Limit
Breaks each for levels 1-3, and one level 4 Limit Break.
Tifa acquires Limit Breaks the same way as Cloud & the others do, except
that hers work differently. When Tifa gets a limit break in battle, she
will attempt to use all of her currently available Limit Breaks. For
each Limit Break she has available, an extra slot appears on a slot
machine that appears when Tifa executes a Limit Break. If you get a
"miss," Tifa will not execute that particular Limit Break during that
battle. A "hit" means that the corresponding Limit Break will do normal
damage, and a "yeah!" means that it will do additional damage.
Vincent only acquires one Limit Break per level. All of his Limit Breaks
change Vincent into a beast. While Vincent is a beast, his HP doubles
and he can't be controlled, but he has some powerful attacks.
Cait Sith only acquires one level 1 and one level 2 Limit Break.