A lot of people don't know how the process to learn Limit Breaks works, or they may not know how to learn all the Breaks. Basically, the Limit Break mastery process goes like this:
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 eight times.
Level 2, first Limit Break:
Kill 80 enemies. You don't have to win that battle, you just have to be the one to kill a particular enemy (i.e. give them the finishing blow) 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 is only 60 for Vincent.
Level 2, second Limit Break:
You must use your first Limit Break of that level six 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 four 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.
When you're trying to learn first-grade Limit Breaks, try giving your character the Slash All materia, the Ultima spell, or some attack spell and the All materia and finding an area where you can encounter many, easy to kill enemies at once. As for second-grade Limit Breaks, use a 'Hyper' on the character who's Limit Break you're trying to get, and give them the Cover materia so that they'll take more damage. If you're really desperate to learn a Limit Break, give them a weak bracelet (like Wizard Bracelet), and a weak weapon, so that they can Limit Break repeatedly without killing the enemy.
Roger Christman has this to say about Limit Breaks:
In other words, my method listed here won't for sure get you all the limit breaks exactly. So if someone can correct me (and has proof that they are correct), let me know and I'll make corrections.
THE EXTREME LIMIT BREAK MANUALS' LOCATIONS ARE LISTED HERE:
CLOUD STRIFE
You must earn a varying number of battle points at the Gold Saucer in one visit. You can then trade them in for the manual.
BARRET WALLACE
Get it from a lady wearing a hat in one of the buildings in North Corel after the train ride in Disc 2.
TIFA LOCKHEART
Go to Nibelheim in Disc 2 and play the 'Highwind' theme on the piano with Tifa in your party.
AERIS GAINSBOROUGH
Get xx wins in battle (where xx are two matching digits, be it 88, 100 or 255). Then go to the cave where the sleeping man is and he'll give you the Mythril. Head to the blacksmith's house by the Gold Saucer, give him the Mythril, then climb up the stairs and examine the metal lid at the end of the walkway. Once you get the Highwind, doing this is a snap. But Aeris will have already left you by then. So, to get her manual while she's still in your party in Disc 1, do the following:
Hop on the Gold Saucer Buggy. If you haven't gotten it yet, keep playing until you've finished the Gold Saucer area. Then pilot the Buggy into Costa Del Sol. You'll enter the town as if you had walked into it. Go to the harbor and talk to the sailor by the boat. Choose the top option to pay 100 gil and you can then walk along the dock and enter a ship, which will return to Junon Town. Although there's not much new here, you will get a different reaction from most people because you're no longer dressed up as a guard. Anyway, once you exit the city (the fastest way is to take the helicopter--refer to the end of section 4e for details), you'll find that you've brought the Buggy with you! From here, it's a simple matter of heading towards the river, crossing it at the ford below the waterfall, then heading up to the cave where the sleeping man is. And after getting the Mythril, you have to go all the way back to the Eastern Continent so you can reach the blacksmith's house. Well, don't say I didn't warn you!
RED XIII
Go to Nibelheim and unlock the safe in the right wing of Sephiroth's Mansion (use the code in section 8 if you don't know how). You'll receive Red 13's manual from the boss when you defeat him.
YUFFIE KISARAGI
Return to Wutai after completing Yuffie's sub-quest. Head to the pagoda and beat the five bosses within--she'll get her manual once Godo is defeated in battle.
VINCENT VALENTINE
Once you have the submarine or a Green, Black, or Gold Chocobo, go to the waterfall in the middle of the Western Continent. If you enter the waterfall from the left side with Vincent in your party, he'll meet a girl from his past (Rukuresshia). You have to visit the falls twice, once during Disc 2 and again before raiding Midgar City; it's during your second visit that you'll get his manual.
CID HIGHWIND
After you get a submarine, go underwater and search for the wrecked plane near the southern part of the World Map, between the Western and Eastern continents. His manual is in one of the treasure chests inside the plane.
Continue to Cloud Strife's Limit Breaks