You'll find him in the World of Ruin. To get him, you have to beat Deathgaze while flying.

You must finsh the game once.

Finish the game.

Kill 8 Dragons in the World of Ruin.

To have Gilgamish as an Esper in the World of Ruin, accumulate 500,000 gil, go to the Jidoor auction. The auctioneer will display a sword called Excalipoor and you must keep bidding until 500,000 gil (it will take a few or many more revisits until the auctioneer actually display the item for bid).

After buying the weapon go to Dragon's Neck Coliseum, located north from Kohlingen. Go straight to the guy who asks if you care to fight in the coliseum, in which obviously you should say yes. He will asks what weapon you would bet on and that's where the Excalipoor comes in. Bet the weapon, choose your fighter and your first opponent is an Onion Mage.

After winning the match, a mysterious character threatens to take away the item. That is Gilgamish. The mid 40s is a good level to muscle through. Have Sabin in your party to throw in the Phantom/Bum Rush followed by Gogo's mimic to duplicate Sabin's attack.

Defeat Gilgamish, you will rewarded a relic item called Merit award, an item which lets the wearer equip any weapons and armors, and the esper: Gilgamish. The spells to learn from Gilgamish are Quick and Valor.