You can and will learn most of the character’s Limit Breaks throughout the regular course of the game. Specifically, characters can learn their Level 1, Level 2 and Level 3 Limit Breaks automatically as soon as they meet certain qualifiers (check out the Learning Limit Breaks section in the Tips and Tricks area for more information).
Characters will need to acquire a special item though in order to learn their Level 4 Limit Break. As soon as they have learned both Level 3 Limit Breaks they can use the item to learn the Level 4 Limit Break and begin using it. Note that there is only one Level 4 Limit Break, not 2 separate Limit Breaks like in the other levels.
Details for where to find the items required to teach characters their Level 4 Limit Break are included below:
You can obtain Omnislash at the Battle Square Arena in Gold Saucer. You need to obtain approximately 32,000 BP at the end of the game (or you can try and get 51,600 points after you have obtained the Tiny Bronco). Check out the Battle Square side quest section for more information about how to earn points in the Battle Square arena.
Omnislash is one of the most powerful attacks in the game. You will automatically get to use it at the end of the game during the final battle against Sephiroth, but obtaining it for use against some of the more difficult encounters in the game (including Emerald Weapon and Ruby Weapon) is critical.
Cloud will attack a total of 15 times with each attack automatically being a critical hit. Each attack hits for 75% damage but because of the critical hits this will most likely result in each attack hitting for 9,999 damage. The only attack that comes close in comparison is 
Knights of the Round Materia.
Tifa’s Level 4 Limit Break is called Final Heaven and it can be obtained anytime after Cloud has rejoined your party (after you pick him up in Mideel on Disc 2). You have to return to Tifa’s house and play the piano up in her room. Check out the returning to Nibelheim side quest section for more information.
Final Heaven adds another attack to Tifa’s Limit Break combo which will deal the highest amount of damage at the end of the chain to a single target.
Barret’s Level 4 Limit Break is one of the more easy ones to obtain. You can pick it up after you have completed the Huge Materia on the train side quest in Mount Corel. Return to North Corel and speak to the woman in the Inn to receive the item.
Catastrophe hits 10 times for 125% of normal attack damage. This makes it a fairly strong attack but it does not compare to Omnislash or Highwind.
Red XIII’s Level 4 Limit Break can be picked up during the side quest to obtain Vincent as a playable character. You will have to return to Nibelheim and open the safe in the Shinra Mansion and defeat Lost Number. Check out the Secret / Optional Characters section for more information.
Cosmo Memory does a ton of damage, however, it will likely be capped at 9,999 for a single hit. It will hit all enemies on the battle field but is still fairly weak when compared to Level 4 Limit Breaks that hit 10+ times.
Cid’s Level 4 Limit Break is found on the Sunken Gelnika. You can reach the Sunken Gelnika after you have completed the submarine mini-game and destroyed the red submarine. Check out the Sunken Gelnika side quest section for more information on where to find the Sunken Gelnika and how to navigate through it and pick up all of the items.
Cid’s Level 4 Limit Break is named after the airship which he calls into battle to unleash missiles upon the enemy. Each attack deals 11/16 regular damage but can hit up to 18 times making it one of the strongest attacks in the game (even stronger than Omnislash).
Yuffie’s Level 4 Limit Break is obtained during a side quest in Wutai, Yuffie’s home town. Doing this side quest before disc 3 means that you’re going to have all of your Materia stolen by Yuffie. Click on the link for instructions on how to get your Materia back from her. After you get all of your Materia back (or if you’re on disc 3 and Yuffie no longer steals your Materia) you can complete the Wutai Pagoda side quest to obtain All Creation.
All Creation will hit all of the enemies on the battle field for 8 times regular damage - again, this will likely cap out at 9,999 HP making it a fairly weak attack overall (when compared to Omnislash or Highwind).
Vincent’s Level 4 Limit Break can be found in Lucrecia’s Crystal Cave. You need to have access to the submarine (and completed the Underwater Reactor section of the game). Check out the Lucrecia’s Crystal Cave side quest section for more information on where to find the cave and how to obtain Vincent’s Level 4 Limit Break and his Ultimate Weapon, Death Penalty.
Vincent will transform into Chaos - a powerful demon-type creature. His physical attack will do 250% damage to all enemies and he can potentially instantly kill enemies using a move called “Satan Slam”. The problem with this move is that you lose control of Vincent for the remainder of the battle.
Note: Cait Sith does not have a Level 4 Limit Break.
