First - ride around till Stygian stays on the surface, use the platform he was tied up to to help keep a little distance, and use the charge ability if he gets to close.
Once he is staying on the surface, ride along side his metal face mask, and hack and slash.
Repeat till the mask comes off.
Make sure you have mercy equiped
Ride around again till Stygian shoots straight up then some small white worms will chase you.
Lock on and just keep firing with Mercy, until they go away and Stygian appears.
When he appears he should try to take a lunge at you, use charge to move out of the way, lock on , shoot, ride.
Second Phase will be repeated a few time, each time adding more white worms, and Stygian keeping up with your riding a little easier. Just make sure you have Ruin's charge available/refilled as much as possible.
After a while of repeating second phase Stygian will slump forward, ride up to him (stay on the horse) and press B for execution.
Watch cutscene and enjoy your achievement