In order to learn a party member's level 4 limit break, that party member must already have learnt all of their level 1, 2 and 3 limit breaks. Then you can use the item for the level 4 limit break on that party member from the Item section of the menu.
You learn each party member's level 1, level 2 and level 3 limit breaks by defeating enemies with that party member and by using the party member's first limit break at each of level 1, 2 and 3 a certain number of times. So for example, have a look at Cloud.
Cloud's Limit Breaks
Cross Slash - use Braver several times.
Blade Beam - defeat a certain number of enemies using Cloud.
Climhazzard - use Blade Beam several times.
Meteorain - defeat a higher number of enemies using Cloud.
Finishing Touch - use Meteorain several times.
Omnislash - use "Omnislash" item on Cloud.
Vincent and Cait Sith have fewer limit breaks total - with these 2 party members you just have to defeat a certain number of enemies. Cait Sith has one level 1 and one level 2 limit breaks, while Vincent has one level 1, one level 2, one level 3 and one level 4.
I find that party members' level 4 limit breaks are not necessarily their most useful limit breaks. For example, both Yuffie's and Red XIII's level 4 limit breaks are basically just like having a Slash-All or a Mega All materia equipped and doing a Slash-All attack - meaning these limit breaks are redundant once you've found the Slash-All or Mega-All materia (full materia guide here). For Yuffie, I prefer her "Doom of the Living" (second level 3) limit break as it deals 15 continuous hits. With Red XIII, I'd also rather use one of his level 3 limit breaks - either "Howling Moon" to put him into a powerful frenzy, or "Earth Rave" for 5 continuous hits.
With Barret, you're better off using "Ungarmax" ("Angermax") as it deals 18 continuous hits rather than the 10 continuous hits of "Catastrophe", as long as you have enough high AP materia in his Missing Score weapon to ensure you can deal out 9999 HP damage with each of the hits (more on the Missing Score). Thanks to a Facebook fan who pointed this out to me.
Vincent also effectively uses the equivalent of either Slash-All or Flash when in level 4 "Chaos" mode - I prefer to use him in level 3 "Hellmasker" mode where he will either do a few continuous hits or a status attack.
With Cloud, Tifa and Cid I recommend using the level 4 limit breaks - in Cloud's and Cid's case, you're dealing a heavy continuous attack ("Omnislash" or "Highwind"), while Tifa will deal up to 7 continuous attacks at limit level 4 ("Final Heaven" being potentially the last part). With Aeris, I think it's a tough call between her level 4 "Great Gospel" and her level 2 "Fury Brand". Great Gospel will fully heal your party as well as granting "Peerless" status (temporary invincibility) - but Fury Brand will fill both of Aeris' allies' limit breaks, which is wildcard in potentially a very powerful way.
Continuous attacks are so useful because they effectively allow you to deal more than 9999 HP damage per enemy, which is the maximum for a "normal" attack. For example, if Yuffie uses "All Creation" she can deal up to 9999 HP damage to each enemy in a group of enemies; but if she uses "Doom of the Living" she can potentially deal 9999 HP damage 15 times..!
Cait Sith only has 2 limit breaks - either level 1 "Dice" or level 2 "Slots". Although "Slots" can randomly kill ANY enemy in the game, it can also randomly give an instant Game Over. As such, I stick to Dice, which is also random but not as unpredictably.
Limit Breaks Guide
Enemy Skill Guide
Final Fantasy VII Remake Guide
List of Every FF7 Title
Other FF7 Websites