Complete chapter 09 and reach Chapter 10
To do this you must work your way to the checkpoint at the end of the bridge access. Once you reach the bridge you face a Fal'Cie by the name of Barthandelus.
Preparation
Items: Librascope x1
Party: From left to right on menu screen, Lightning, Hope, Fang.
Paradigms: Relentless Assault (RAV-RAV-COM)
Solidarity (COM-MED-SEN)
Diversity (RAV-MED-COM)
Bully (COM-SYN-SAB)
Hero's Charge (MED-SYN-COM)
Accessories: If you have not been collecting treasure spheres on your travels you might need to buy from the shop some of these accessories.
Lightning - Tungsten Bangles x2
Hope - Shaman's Mark, Gold Bangle (the Gold bangle can be found in a treasure sphere in the bridge access area on a platform to right of the platform the save checkpoint is just before you fight the boss at the end of this chapter)
Fang - Brawlers Wristband
Crystarium: It would be wise to make sure you have reached the 3rd level role level crystals for Lightning, Hope, and Fang. If you have not done so it would probably be best to Grind for CP in the Bridge Access area.
The Fight:
You will Be facing a Fal'Cie named Barthandelus who has an insane amount of HP for this point of the game and has the capabilities to wipe out your entire team with one attack if your not careful.
as soon as the battle starts you will notice that besides the main head there are also 4 additional upside down heads. each one of these heads must be destroyed to reduce overall strength and frequency of attacks also destroying the 4 pieces of armor causes lose of damage resistance and blanket immunity to status ailments. Start by switching to Bully paradigm to bestow protective buffs. While Hope is buffing up your party use librascope to discern the elemental weaknesses of each upside head. Once hope is finished quickly switch to Diversity to both Heal and attack each of the heads. Whenever you feel safe enough, switch to Relentless Assault to speed things along. Once you have defeated the outer armor or shell you can start attacking the main target. Switch between Relentless Assault and Diversity as needed until you are able to stagger. Once staggered the you have 2 choices, you summon Odin and knock out a chunk of health now using auto Gestalt or save him for a latter time and continue to use Relentless Assault. After the first Stagger Barthandelus will use an attack called Destrudo. This attack has the capabilities to wipe out your entire party if your not paying attention and will reset the stagger bar even if already staggered. When you see the prompt on the screen that says Destrudo quickly switch to the Hero's Charge paradigm. With the Hero's Charge paradigm you have 25 seconds to manually heal every one's HP so that every one is in the green (preferably full health) once every one is of good health immediately switch to Solidarity to reduce the amount of damage you take from this attack also you should see Hope heal Lightning during her greatest need after the attack. Once Lightning is healed to at least the yellow switch to diversity to further heal and do damage to the boss switching to Relentless Assault to speed things along when felling safe enough.
Towards the end Barthandelus may or may not cast the incurable Doom status on the party leader but don't freak out, the doom status actually allows you time to finish the fight.
Rinse and repeat the process until the boss is defeated.
GOOD LUCK!
NOTE: Depending on how quick you are with paradigm shifts and how aggressive you are with your attacks, the Doom status will ultimately decide your fate the second it is cast.
I myself had to fight Barthandelus a second time because of the Doom status that killed me. The second time around however Barthandelus did not cast doom.
Watch a couple cutscenes then the cheivo should pop.