To learn Level 4 limit breaks you'll need to find the manuals. Here's how to find them.

Cloud: Trade 32,000 battle points at the Golden Saucer.

Tifa: Go to Nibel area (Tifa's house) and play these piano notes: X, Square, Triangle, L1/R1 + Triangle, L1/R1 + Square, X, Square, Triangle, L1/R1 + X, O, X, Square, X. Then press START and search the music sheet to find Tifa's Limit break.

Barret: A lady will give it to you in North Coral.

Aeris: Southeast of Midgar there is a small cave with a man living there. Talk to him when you have battled a certain number of battles that end in the same number, such as 122, 133, 144. He will then give you a Mithril orb. Take the Mithril to the Blacksmith's house on the south peninsula near Cosmo Canyon. He will trade you for the limit break manual for the Mithril, or you can choose to have a gold bracelet.

Red XIII: In the Nibel area there is a mansion and on the second floor of this mansion there is a safe. To crack this code turn right 36, left 10, right 59, and right 97. Then the safe will open to reveal a miniboss.Kill the miniboss and you will recieve Red VIII's Manual.

Vincent: Land near the waterfall and use the green, black, gold chocobo or the submarine to pass the mountain. Then enter the waterfall. (You must have Vincent in your group and must be done first in disc 2) go forward and it will show Vincent's past. Then leave. Return if you are already in disc three and you'll get not only the manual but a weapon of Vincent.

Cid: Cid's manual is in one of the chests in the sunken plane.

Yuffie: Go to Wutai and enter the tower with yuffie and let yuffie beat all the bosses including her father and you'll get a manual and Leviathan materia.