There are two things you need to do to learn limit breaks.
To get the second limit break for a level (ie. Climahazard) you have to use the first level one a certain amount of times.

To get to a new limit break level (ie. Climahazard->Blade Beam) you need to kill a certain amount of enemies.

For killing a ton of enemies I usually like to go to the mithril mines and run around the screen where you first enter from the Midgar side. Enemies come in a minimum of 2 at a time, 4-5 is the most common. And they are easily wiped out by very inexpensive spells.

To get lots of limit breaks I recommend equipping the characters with as much HP reducing materia (all your most powerful magic) to get their HP as low as possible, and equip them with the worst armor you have (wizard bracelet is perfect). Also Fury is a must. Then just find any manipulatable enemy with a decently strong attack and give'er. You can lower your HP bellow 1000 even late in the game, so almost anything will work.