Unlike your SSS playthrough as Dante, you'll want to keep up one of Vergil's devil trigger attacks for pretty much the entirety of the campaign. I highly recommend going the Spiral Sword route as this doesn't do a ton of damage, allowing you to make longer chains, but it is also VERY good at stunning enemies that come close to you if you aren't able to react in time, also saving your chain. The little extra boost in damage also helps you speed through the levels at just the right pace and is perfect for helping you cut down multiple flying enemies at once while you attack one.
Other than that, you will also want to change your abilities up a bit, focusing on Angel Mode attacks as they are all lightning fast and deal damage to multiple enemies, but you'll also want to mix in some standard attacks and most importantly you'll want to do some perfectly timed perfect slices and atomics, using your standard Yamato and demon mode respectively. Atomic is especially effective at keeping multiple heavy enemies (like the chainsaw guys or even an imprisoner) relatively stunlocked when they work alongside your spiral sword, allowing you to pretty efficiently clear even the toughest enemies without losing your combo.
As you can tell, the key is to keep your combo going, and the best way to do that in this DLC especially is to avoid extended periods of floating in open space. The less often you fall off a ledge, the less often you will break your combo, and it can be very easy to fall off (and even be launched off..kind of glitchy) of ledges in this DLC.
This is considerably easier than the similar achievement in the full game, even though the fight with Shadow Vergil may get kind of tricky, the style points requirement is rather low for that fight.