You can only learn Fourth Limit Breaks by learning all previous three limit breaks and finding the Fourth Limit Break in the game, as they can not be learnt.
Aeris
(needs to be ported to the wiki!)
You can only use this limit in Disc 1. First of head over to Costa Del Sol in the buggy given to you by Dio, and drive on into the city, and take the boat over to Junion. The car shall follow you over to the other continent. Head North-West where you come to a stream with a shallow part, leading to a area you haven't been before, just south of Midgar, but on the other side of the mountain chain. You will see a cave there with a sleeping man inside. go inside, and he will tell you random things, but what comes in handy is the info about how many battles you have escaped from. You need Mythril, and this man will wake up giving you some. However, he only gives you some, if the number he tells you has the last two digits the same. For example, 266, 344. When you have gotten a hold of some Mythril, go back over to Costa Del Sol with the buggy, and head directly South until you come to the river, and cross that river. Right after you cross the little river, take a sharp right turn, and head west until you see a little house almost at the edge of the continent. Inside you will find a blacksmith who needs Mythril. Give him the Mythril. He will then reward you, and you get a choice of getting the big or small box. Choose the small box, and bring it to him. You now have Aeris Fourth Limit.
Barret
(needs to be ported to the wiki!)
In disc 2, after you have found the Huge materia in the Reactor on the train north of North Corel, you can obtain this item from a lady inside one of the houses. You will only get this Limit, if you managed to stop the train the Huge Materia was in.
Cait Sith
(needs to be ported to the wiki!)
Cait Sith does not have a Fourth Limit Break.
Cid
(needs to be ported to the wiki!)
You can get this limit break when you retrieve the Submarine. When you have the Submarine, you need to head for the Shinra plane, Gelinica. It should be at the bottum of the sea, north of Costa Del Sol. Scan the sea walls/bottom in that area until you find a pile of rubble shaped as a "plane". You will find his limit break through the door, north from the first room you come in as you enter the plane. I recomend you escape all battles in that room if you're not in a good shape as this is where many of the worst enemies can be located.
Cloud
(needs to be ported to the wiki!)
Getting Clouds 4th Limit Break takes a lot of time and patience. You also need a great character with great stats. Gold Saucer's Battle Arena is the place to to get this item. Have everything you will need as you can not leave the battle arena once this starts (if you wish to get the Fourth Limit Break), as you will loose all your Battle Points. You can check you battle points (BP) on a screen mounted in the roof, close to the stairs, which will lead you out of the Battle arena on each side. To get BP, you need to battle in the battle arena. If you loose a match, you only loose the points you gathered in that round, and not BP from previous battle rounds. There is 7 battles per round, and for each round, you loose a ability. It can be anything from loosing all you magic materia, to half HP, and to be turned into a frog. You can control what the reels (which decide what status effect you migth get) land on, but it's not easy, so keep a close eye on it. For each fight in a round, you get a choice to back out. I would recomend to back out when you absolutely sure that you can't win the rest of the fight's. You will get extremely few points by leaving the fight unfinished, but many have found that I only needed a few point's to get Omnislash, where you need to start a new battle just to get those extra few points. Before Aeris dies, the BP that you need to gain lies on 64 000 BP, but if you wait until disc 2, you can get it much quicker and effortless at 32 000 BP. This limit break is the most devasting attack you can command in the game.
Red XIII
(needs to be ported to the wiki!)
You can get this Limit on any disc. Go to Nibleheim and enter the big Shinra Mansion. Head for the safe which is above the ground floor to the left. You will see a safe up against the north wall. You need to punch in a code before the time runs out, but be careful not to pass the numbers, because if you need to press the button to go back to the number you passed, it won't be valid, so just press in random numbers and start over again if you mess up the first time. The code is:
Right 36, Left 10, Right 59, Right 97.
You will get a summon and a key to a secret door (used to get Vincent) as well as Red XIII's Fourth Limit Break.
Tifa
(needs to be ported to the wiki!)
You can get this item in Disc 2, when Cloud and Tifa review Clouds past in Mideel. Return to Nibelheim and go into Tifa's house where the Piano is in the 2nd floor. There you can play the piano, and you must play a very familiar tune. Press the following keys:
X, , ^, R1 + ^, R1 + , X, , ^, R1 + X, o, X, , ^.
If you do it correctly, 2 Gil will be given to you, until you eventually find a letter from Zagan (Tifa's Mentor) which contains her final limit.
Vincent
(needs to be ported to the wiki!)
In disc 2, if you either have the mountain climbing Chocobo (or better), or the submarine, you can get this one. If you have the chocobo, head over where the Gold Saucer is on the world map, and head just about directly west. Here you will find a little lake surrounded by mountains, and you should clearly see a cave there. Just walk on over there, get of the chocobo, get Vincent on your team, and then head on inside the cave. If you get a piece of Vincent's past shown before you, you're on the right track. Leave, and after a special event, you should return where the cave should be empty, but at the far north in the room, you should find not only Vincent's final limit, but his ultimate weapon as well.
Yuffie
(needs to be ported to the wiki!)
First you must have Yuffie as a party character. Read here to learn how to do this.
Then you must head to Wutai (her hometown) and partake in a mini quest where you must find Yuffie when she steals all your materia. Once you have found her and she joins your party again, head to the large tower you can see in the North-West of your screen (when you're in the main part of the town). equip Yuffie so she is suitable to battle and enter the tower. You must now fight four monsters and then a fifth monster to receive Leviathan and her Fourth Limit Break.