To unlock Ultima HM you need to do a quest called "The Minstrel's Ballad: Ultima's Bane"

You can start the questline by talking to Alys at Mor Dhona (x22,y8)

Here's the video of how to do it


This is an explanation of the strategy (it's a lot to read but it will save you hours of failed tries, read it as you watch the video and it'll be easier to understand):

First of all you need 2 tanks, and they need to be switching OT (Off-Tank) and MT(Main-Tank) roles every time they get 3 buffs on them (like in T2). When one of them gets 5 buffs its instant death, so just switch at 3 in case something gets messy, these buffs do not make you get less or more damage you just need to not have 5 at any time.

Second , white mage must have Ultima as his Focus Target (shift + F) so they can see when he's doing the skill "Tank Purge" which is a strong AoE so they get medica ready after he does it. everyone should gather behind Ultima so they can get healed by this, except for the current MT.

Now for the rotation:

Phase 1: Garuda

(0:25) Garuda appears in the middle of the platform, watch where she's facing because she'll do an AoE which covers all her front, you need to get behind her asap, at the same time Ultima will do an attack that knocks you back, make sure you are positioned so Ultima doesn't knock you in front of Garuda. (MT must avoid this attack too)

(0:53) Right after you'll see a blue circle around the platform , like the one on Garuda's Trial, you need to get to the corner of the middle circle so you don't get hit by her AoE, and right after the circle banishes you need to run away from that inner circle because Titan will fall there doing a huge AoE, just like in Titan HM the closer you are to the corner of the platform the less you get dmgd by, so get as close to the edge of the platform as you can.

Phase 2: Titan

(1:36) Basically the ifrit's plumes strategy, imagine an inner circle, you need to be in the corner of that circle at all times, Ifrit will spawn plumes all around the outer circle and RIGHT after that, Titan will spawn his plumes in the inner circle so you must move out quickly.

Phase 3: Ifrit

Healers seem to have issues with this phase but its really easy to prevent all the moves, just keep tanks with regen and run a lot.

(2:30) First of all Ifrit's gonna start spawning Eruptions over two members, and you cannot stun them like Ifrit HM, you just keep running around the corner of the map, while the members who didn't get target by them stay in the middle.

(2:55) After this you'll see 4 plumes spawn on each corner of the platform, obviously you need to avoid them but most importantly you need to watch around for Ifrit because he's gonna do his dash move while spawning another set of plumes in a line. He'll do this twice in a row so be ready for both. Apparently he always spawns and dashes to the same directions, I have yet to confirm this as I've never paid attention to the compass on this fight.

Phase 4: Bombs 1

(3:56) Four bombs appear linked in pairs , each bomb of the link will move slowly together, if you touch one, both linked bombs will explode, if you don't touch any and they get together, they'll AoE everyone for around 10k dmg to each person.
There's two way's of doing this, either everyone gathers around one person and they make them explode one by one or everyone gets to a corner away from range and OT eats them while the healers keep him full of hp. in the video the OT makes one of the links explode while a group of other players make the other link explode together.
I usually mark myself and have everyone follow me exploding them, easier if everyone gets in the center right before this phase so everyone's already gathered and the just move a little to trigger the bombs, if all the party members are together including the OT (not the mt), dmg should be very low, but still ready medicas just in case.

He'll also start doing a move where he shows a green line on the floor and obviously you have to avoid it, he does it three times in a row.

Phase 5: Laser 1

(4:40) Five Magitek bits will appear in the center and they'll shoot lasers all over the place, this is way easier if everyone focus on one at a time so all lasers point to the same position and you can move around, before you get to kill them all, some missiles will crash on the floor, they'll give you plenty of time to evade them and will also show you the range of AoE on the floor as red circles, the closer you are to the center of the circle the higher you get hit (5:00) (this was one Ultima NM too)

Phase 6: Bombs 2

(6:05)This time its going to be 8 bombs, same strategy as Bombs 1, except if your healers are not fast enough you might not being able to do it with just the OT detonating them, as you gotta do it quicker. Better if you do the everyone detonates them together strategy (like they do in the video)

Phase 7: Laser 2

(6:30) Bits will spawn again shooting lasers all over, this time they'll be all around the edge of the platform, better to start on the one at 6 o'clock and everyone move together killing them counter clockwise. (6:46) When the missiles are coming (you see red/purple circles on the floor), everyone get togheter in the middle asap, as they all will spawn on the edge, healers keep curing because lasers will keep shooting and ready some medicas too.

Phase 8: Bombs 3

This is where most people wipe on pugs and its actually very easy , but all 8 members gotta know their role.
(8:04)This time the bombs spawn very close to each other so you don't have time to trigger them pair by pair. There's a couple of different ways to go through this phase.
The one on the video and the one I like the most is get close to the tank and right before they spawn have him LB so everyone gets defense buff and move around detonating all bombs quick. None of the links can get together or its a wipe.

Another way to do this is to have the MT and one healer away from all the bombs, everyone makes them blow up dieing with them then healer LBs, sounds weird but it works, as long as the tank doesn't die or the hate will be all over the place and only works if you have 3 bars which is not that hard at this point unless you have really high dps.

The third way to do this is to make groups of 3 people and each group detonates a different link, everyone needs to be synced to do this and plan it before the fight. This is the hardest to pull off because of one single reason: people don't like to listen and they always rush in without a care in the world :)

After this you just have to burn Ultima down, tank and spank !

Summary

Tanks switch hate at 3 buffs.
Whms get Medica ready when Ultima starts casting Tank Purge.

Phase 1: Garuda

(0:25) Get behind Garuda when she appears. Stay in an inner circle and get out to the edge of the platform after Garuda's AoE.

Phase 2: Titan

(1:36)Stay in an inner circle then go out after outer plumes explode.

Phase 3: Ifrit

(2:30) Evade Eruptions. Evade Ifrit's Dash and Plumes, make sure you look for him after the first set of plumes so you'll know where he's dashing to.

Phase 4 & 6: Bombs 1 & 2

(3:56) & (6:05) Everyone but MT move together around detonating each pair of bombs. 1 Healer can stay still, depending on skill.

Phase 5 & 7: Lasers 1 & 2

(4:40) & (6:30) Kill lasers starting at 6 o'clock then one by one clock counter wise. Reduce Missiles' damage staying away from the red/purple circles that appear on the floor. On Phase 7 you can avoid the must of the damage by having everyone stay in the middle.

Phase 8: Bombs 3

(8:04) Have the OT LB near everyone right before this phase and when the bombs appear spread in groups to detonate them all. Burn Ultima.