the other method on how to do this works but it is to long if you want to get SSS on Nephilm difficulty.
The method in the video is high risk depending on your play style but it suited me perfectly.
Use rebellion as the primary weapon its fast and dose a fair amount of damage.if you feel confident you can combo in osiris to increase your style meter. secondly use revenant shotgun to clear the ghost blades (as its put in the vid)
Now as the fight starts run up to Virgil and just hack away with Rebellion either you trigger a cut scene quickly or he do a charged sprint attack at you dodge this and use the Arbiter pull and do some more damage with rebellion. after about 2 cut scenes Virgil will starts to use more attacks similar to your aerial flush attack. this can be countered in one of 3 ways by a simple dodge, or jump above it, and finally a bit more tricky defect it back at him. which ever method you use use the arbiter pull to bring him in close and use rebellion again.
the final stage of the fight is were Virgil spawns a shadow clone of himself. use concentrate on the original until Virgil is kneeing on the ground.
the use your devil trigger hit him once to start the final cut scene.
if you have no devil trigger energy hit the clone. this gives you purple orbs to fill you trigger.
I have done two runs using this method both clocking in just over 4:33 for each ( done on nephilm difficulty)
if you have any questions about this method I will try and answer them a quickly as possible.
Most of the method is shown in this video i found on Youtube. I didn't make this video the credit goes to it's creator.