Credit goes to SinestroX on steam.
This is a Missable achievement. Do this before going to the Temple of the Ancients, because after this part Aeris will leave your group and you will not be able to make her learn the Limit Break. This achievement can only be obtained when Aeris LEARN the Limit Break.
1) Learn all the Limit Breaks of Aeris up to Level 3 before going to the Temple of the Ancients. This can easily be done on the first Disc (First part of the game).
2) Once you get the Tiny Bronco, go back to Nibelheim where the buggy was left, then take the buggy all the way to Costa del Sol (enter Costa del Sol with the Buggy) where you can pay 100 gil to get on the cargo ship. Once the ship was taken to Junon, you can take the Air Taxi out of Junon by asking the guy in white sailor outfit outside the cargo ship area and then you'll end up on the world map, with the Buggy.
3) Now go to the Sleeping Old Man who is found in a cave directly east of Junon. According to @TheGreatKoala he doesn't show up until you complete Rocket Town. You will need to cross the river by using the buggy in order to reach him. He'll say how many times you've fought, he'll mention something about Huge Materia, he'll say how many times you've escaped. But he will only wake up and give you some Mythril when the last two digits of the numbers of battles you have fought are the same. If he gives you the Bolt Ring, come out and fight until the last two digits become the same again, talk to him again and he will give you some Mythril.
4) Once you've all the Limit Breaks of Aeris up to Level 3 and the Mythril, take the buggy (or Tiny Bronco) over to the weapon seller, who lives in a house right across the river from the Gold Saucer. You can not miss it. Talk to him, choose the third option, give him the Mythril and he'll let you open one of his two boxes. The large box contains just a regular item, so DO NOT OPEN IT! There is a small box on the second floor which is attached to the wall. In it, you'll find Aeris' Limit Break Level 4 Great Gospel.