You cannot fight Vergil head on. He is programmed to deflect/parry off anything and everything and is vulnerable only during specific openings when or after he attacks.

Vergil can be launched via launch attacks during any of his openings, but he can still break away/parry you off very quickly.

Summoned Swords:
The summoned swords move/attack on their own accord - they will not break/disappear even if you land a hit on Vergil.  The easiest way to avoid them altogether is to use Aquila's regular attacks (/\ button) or Osiris' Shredder (O button), which will break all his swords in a single attack. Vergil is quick to move on to his next move once his swords are gone so be on the alert.

Rapid Slash:
Vergil charges for a moment before he dashes forward, leaving an invisible trail of slashes behind him. The slashes can hit you even if you're in midair so be sure to stay out of his line or stay in midair (keep shooting or do attacks that stall you in midair like Arbiter's Flush). There's a short opening for you immediately after he lands.

Stomp:
This move's pretty fast. Be on the alert to evade once you see him teleport above you. Angel/demon evade is better than jumping. There's a short opening for you immediately after he lands. The easiest move to get in on him after evading this is Aquila's Big Slick.

Flush:
Vergil does this in midair only once, but up to three times on ground. In midair, you can either deflect the sword at him (requires timing), or you can quickly evade it and then immediately angel lift over to him for an opening. This is by far the most common opening.

Killer Bee:
Vergil dives kick at you from midair. It locks onto you so running off in a different direction does not work - you must evade it with either angel/demon evade or a well-timed jump. There's a short opening for you immediately after he lands.

Judgment Cut:
The best and easiest way to avoid this is to shoot a Kablooey dart while Vergil's in his stance and detonate it immediately. This will cancel/interrupt his attack. Alternatively, you can keep running around Vergil (safest with angel glide or angel evade).

Doppelganger:
Vergil will create a clone of himself when his health is down to about 3 bars. In this mode he will slowly regenerate back to about 50% health (assuming that you don't hurt him at all in this time), after which the clone will disappear. In doppelganger mode Vergil takes less damage in general.

Now with two targets it becomes harder to target your attacks/lifts/pulls at Vergil and not the clone. While his openings remain the same, the clone could abruptly interrupt your (ground) attacks if he's close by, and Vergil himself parries you off much sooner.

Once you bring Vergil's health back down to half a bar, he will become immobilized and the doppelganger will reappear. Wait for the doppelganger to move away a fair bit from Vergil, then activate devil trigger. Immediately run over to Vergil and hit him once and the battle will be over. If the doppelganger's not far enough from Vergil, he will shield Vergil no matter what you do.

Last updated: 16 Feb 2013