In order to use the Level 4 Limit Break, you will first need to have learnt all of that character's 6 previous Limit Breaks: 2 at Level 1, 2 at Level 2 and 2 at Level 3.
This can get tricky as it's possible to, for example, learn the first Level 3 Limit Break before you've learnt the second Level 2 one. In order to learn the second Level 2 Break, you would need to stay at Level 2 and use the first Level 2 Break a certain number of times.
Basically, the first Level 2 and Level 3 Breaks are learnt by killing a certain number of enemies. The second Level 1, 2 and 3 Breaks are learnt by using, respectively, the first Level 1, 2 and 3 Breaks a certain number of times. This should provide you with enough information to know what you're doing.
To learn the Level 4 Break, select the Level 4 Item from the menu and use it on the correct character.
N.B. There are 2 exceptions to the above. Vincent only has one Limit Break at each Level, so it's easier with him. Cait Sith only has 2 Limit Breaks total and has no Level 4 Break.
For an excellent article which goes into more detail, see STRATEGYWIKI.
Personally, I find that characters' Level 4 Limit Breaks are not necessarily their most useful Limit Breaks. For example, both Yuffie's and Red XIII's Level 4 Breaks are basically just like having a Slash-All or a Mega All Materia equipped and doing a Slash-All Attack. For Yuffie, I would be using 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 Breaks - either Howling Moon to put him into a powerful frenzy, or Earth Rave for 5 continuous hits.
With Barret, as long as you have enough high AP Materia in his Missing Score Weapon to ensure you can hit for 9999, you're better off using Ungarmax (Angermax) as it deals 18 continuous hits rather than the 10 hits of Catastrophe. 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 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 would recommend using the Level 4 Breaks. With Aeris, I think it's a toss-up between her Level 4 Great Gospel and her Level 2 Fury Brand. Fury Brand will fill both of Aeris' allies' Limit Breaks, which is wildcard in potentially a very powerful way.