Here is the full list of all 96 Final Fantasy V (Old ver.) achievements. It takes around 60-80 hours to unlock all of the achievements on Windows.
You escaped from the final battle.
All four characters mastered the necromancer job.
All four characters mastered every job.
All four characters mastered the berserker job.
All four characters mastered the bard job.
All four characters mastered the gladiator job.
All four characters mastered the chemist job.
All four characters mastered the cannoneer job.
All four characters mastered the dancer job.
You cleared the Cloister of the Dead.
You defeated Neo Shinryu.
You completed the entire Bestiary.
You defeated Omega Mk.II.
You defeated Enuo.
You raised Galuf to level 50.
You raised Bartz to level 50.
You raised Lenna to level 50.
You raised Faris to level 50.
You defeated Shinryu.
You defeated Omega.
"You defeated Neo Exdeath without anyone in your party dying."
You defeated Neo Exdeath.
All four characters mastered the dragoon job.
All four characters mastered the geomancer job.
All four characters mastered the beastmaster job.
All four characters mastered the mystic knight job.
All four characters mastered the mime job.
All four characters mastered the ranger job.
All four characters mastered the knight job.
All four characters mastered the samurai job.
All four characters mastered the black mage job.
All four characters mastered the white mage job.
All four characters mastered the time mage job.
All four characters mastered the blue mage job.
All four characters mastered the monk job.
All four characters mastered the oracle job.
All four characters mastered the thief job.
All four characters mastered the summoner job.
"You defeated Exdeath's servants--assassins from another dimension."
You pinned your hopes on Gilgamesh.
You earned 500,000 gil.
All four characters mastered the red mage job.
You obtained the twelve legendary weapons.
All four characters mastered the ninja job.
You met the dwarves.
You defeated the mime Gogo.
You mastered the piano by playing all the pianos.
"You got the four tablets for unsealing the legendary weapons."
You defeated 2,000 enemies.
You completed 60% of the Bestiary.
You reunited with Ghido in the third world.
You defeated Exdeath.
You defeated Gilgamesh again.
You earned 200,000 gil.
Krile inherited Galuf's power.
You obtained a submarine.
You mastered lupine attack.
You earned 50,000 gil.
You met a moogle.
You defeated 1,000 enemies.
You raised Galuf to level 25.
You defeated Gilgamesh.
You arrived in Galuf's world.
You raised Bartz to level 25.
You raised Lenna to level 25.
You raised Faris to level 25.
You completed 30% of the Bestiary.
You obtained a job from the earth crystal.
You obtained an airship.
You discovered Faris's true identity.
You obtained a job from the fire crystal.
You earned 10,000 gil.
You met Cid.
You obtained a job from the water crystal.
You defeated 300 enemies.
You completed 10% of the Bestiary.
You obtained a wind drake.
You raised Faris to level 10.
You raised Galuf to level 10.
You raised Lenna to level 10.
You raised Bartz to level 10.
You found out Faris is a woman.
You defeated 100 enemies.
You obtained a job from the wind crystal.
You obtained the pirate ship.
You played a new game.
You earned all achievements.
You opened all the treasure chests.
You raised Bartz to level 75.
You raised Bartz to level 99.
You raised Lenna to level 75.
You raised Lenna to level 99.
You raised Galuf to level 75.
You raised Galuf to level 99.
You raised Faris to level 75.
You raised Faris to level 99.