As with DMC, complete each successive difficulty to unlock the next. On the final difficulty (Hell and Hell) you will be dealing with remixed enemies at full health, but you will die in one hit. Again, you will have 3 golden orbs to revive you, basically giving you 3 freebie deaths before a continue. You will also unlock Super Vergil (the outfit is unlocked during mission 6), just as Super Dante in DMC was unlocked, by beating Vergil Must Die. Thus you will have infinite doppleganger and spiral sword for the duration, which will make life easier, alongside Vergil's innate speed which is considerably faster than Dante (primarily because I found his angel mode moves more well rounded).
I recommend you practice perfecting the Atomic skill ( + hold :bbut) as that is able to knock up every ground enemy in the game. For the aerial types, you should practice pulling them in ( + :xbut) and then tapping by itself to jump straight up and slamming + double tap to stomp on them. Repeat this jump-stomp process rapidly to stunlock enemies, and this is especially effective against Wisps. It's also important to master the Perfect Slice as it does considerable damage to very large groups of enemies, lowering the time you need to spend fighting enemies by a considerable margin. Of course, don't forget to keep up Spiral Sword and Doppleganger (who you should keep in angel mode for crowd control - ), and you might as well use Blistering Swords as much as you want to help take down flying enemies quicker.
Tips for Shadow Vergil
Far and away the hardest part of this DLC will be the fight with..yourself..until you figure out the pattern. For the most part you want to be aggressive as far as staying on top of him, as this tends to lower the chances of him using blistering swords, while increasing his chance of doing any of his aerial meteor attacks. Use the standard attacks as your primary damage, using Angel Mode and dodge to sprint around the screen, allowing you to deal instant damage when there's an opening with (Hold for Perfect Slice) being your best combo, and pause being very good as well, although it actually tends to be a bit slower in my experience. Anyway, when the boss pulls back to do one of his aerial attacks, jump straight up once to completely avoid damage and allow you to quickly drop down to attack him while he is defenseless. Pair this opening with your doppleganger ( + , also he seems to be most effective as demon mode so you're attacking at different intervals) to deal immense amounts of damage in this opening. During the intermission phases, where he pulls out of range and uses sword attacks, make sure you just get the dodge timing down, which only requires some practice, and prepare for the next phase where he speeds up a bit, but the same strategy as stage one still applies. By the final stage he will have very little health left, and you should be able to quickly finish him off.
One side note for Shadow Vergil is that you can use Spiral Sword to block a single Perfect Slice attack, and reuse the ability to block the next and so on, but this seems to keep Vergil farther away from you and constantly moving away from you, which can be frustrating.