A SSS Style Rank is unlocked when a good variation of combos is applied without taking any damage. This achievement will of course be easier if most of the upgrades are unlocked. Try to combine all four Angel & Demon weapons and your firearms in the same fight.
Personal tips: The Trinity Smash is an extremely powerful combo that gives a lot of style points (Rebellion + Pause (Arbiter) + ). I'd suggest using this ability when a few enemies are remaining. It's a very slow attack, you don't want other enemies to hit you while you perform your Trinity Smash.
The Tremor attack (Arbiter + ) is also very good to hit many lined up enemies at the same time. Another combo that greatly helps me to rack up a lot of points is the combination of the Buy In, Killer and Cleaver (Aquila + Pause + ) + (Aquila + + + ) + (Osiris + + Pause + + ). Buy In and the Eryx Stomp will help you tremendously -> (Aquila + pause + ) + (Double Jump + Eryx ).
The start of mission 11 on is a good spot to unlock the achievement.