In order to get Tifa's final limit break, you need to unlock the six prior limit breaks and obtain the item that teaches Tifa her final limit break. The way to unlock each limit break differs for each character but are essentially similar. You unlock the second limit of each stage by using the first limit of that stage a bunch of times, and you unlock the next stage by getting a certain number of kills.
Tifa's limit breaks:
Stage 1: Beat Rush, Somersault
Stage 2: Water Kick, Meteodrive
Stage 3: Dolphin Blow, Meteor Kick
Stage 4: Final Heaven
Unlock order:
Use Beat Rush 9 times in order to unlock Somersault.
Get 96 kills with Tifa in order to unlock Water Kick.
Use Water Kick 7 times in order to unlock Meteodrive.
Get 96 kills with Tifa after unlocking Water Kick in order to unlock Dolphin Blow.
Use Dolphin Blow 6 times in order to unlock Meteor Kick
Use the Final Heaven item after unlocking the 6 previous limit breaks in order to unlock Final Heaven
In order to obtain the Final Heaven item you need to play a certain tune on the piano in Tifa's house in Nibelheim. However, you have to wait until Cloud has returned to the party after falling into the Lifestream. If you enter the correct tune before this time, you will not be able to get the item. You must also have Tifa in the party when playing the piano in order to get the item.
The tune is as follows:
[Cancel], [Switch], [Menu], [PAGE-UP + Menu], [PAGE-UP + Switch], [Cancel], [Switch], [Menu], [PAGE-UP + Cancel], [OK], [Cancel], [Switch], [Cancel].
(Do, Re, Mi, Ti, La, Do, Re, Mi, So, Fa, Do, Re Do.)