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