Once you have defeated all of the preamble bosses to Ultimecia and regained your abilities, you need to return to the Gallery and ring the bell at the bottom of the staircase. You will then have 1 minute to get to the Chapel where Omega will be waiting - you’ll previously have seen a portal in the room when passing it earlier. Simply run into him to engage in the fight.
There are hundreds of guides that will provide strategies for owning him legitimately, but if you purely want the achievement and not the glory, you need to do just a few things:
Firstly, you need a party of three with 9,999hp (you can technically do without, but it’s easier/safer to have them). Ideally, if you want to go on brainless mode, they will also have high strength stats.
Secondly, you need to have no elemental junctions to their attacks due to Omega’s resistance.
Finally, put the game on Triple Speed and Easy Fight Mode and spam the thing with physical attacks until it is dead. In the fight mode you will always be healed after getting hit, so don’t need to worry about anything. It is possible he can still wipe your entire party with an insta-kill attack, so having set up for the Phoenix GF may be helpful too.