Instructions
Use a combination of Synergist and Saboteur paradigms. The Synergist will buff your party up, while your Saboteur will weaken Orphan's defenses. Orphan is strong against Saboteur debuffs so it may take a while.
Switch to Ravager Paradigm to inflict damage and stagger Orphan. Do this after the debuffs are in place, especially deprotect, deshell, and imperil.
After you stagger Orphan with powerful attacks, switch to Medic and Sentinel paradigms to defend against the Fal'cie's Merciless Judgment super attack. Switch quickly as this attack can overwhelm your party. Switch back to the Synergist/Saboteur combo after the barrage.
Repeat the first 2 steps above after he casts Doom. After he casts Doom, time will be against you, and you must defeat Orphan quickly. After deprotect, deshell and imperil are in place, quickly switch to Ravager and stagger Orphan. The last stagger will be his downfall.