1. He usually starts by walking around, and in most of such situations, shooting does not work since he'll deflect them with his kick-ass defensive sword spinning move.
    
2. Typically, Vergil will then teleport towards you and do a sword combo, so dodge it by quickly (double) jumping away from him.
    
3. Vergil is vulnerable every time he finishes a move. After avoiding his attack, take the opportunity to attack him as he sheaths his sword. Close your distance quickly with Stinger if need.
    
4. Vergil will often parry you, and when that happens, (double) jump away immediately. Even then, it's common to get hit while trying to evade. The only way to fully avoid this is to anticipate when he will parry so that you'd know to stop your melee attacks and shoot instead. Vergil always parries after a certain fixed number of hits, and when he is ready to parry, he grunts. Either count the number of hits you've given him, or listen carefully for his grunt, then stop melee-attacking and shoot. For more information, refer to Avoid & Attack strategy #5 on this page.
    
5. Vergil only does 2 different moves after parrying - 1) Rapid Slash (dashes forward and attacks all in the way) 2) Upper slash (2 hit slash). Rapid Slash is very unlikely to hit you if you jump away once parried. Upper Slash is the troublesome one, so try your best to jump away in time, or follow the step above and don't trigger his parry at all.
    
6. After taking about half his life, Vergil starts performing Judgement Cut, a move where he conjures energy waves at you. To avoid them, just run and jump away, also making sure you don't land or jump right into it. This is a good chance for you to counter since he stays stationary as he attacks. Get close to him as you avoid the waves, then use Stinger to stun him and continue on with a combo. A few repeats of the same things mentioned above should finish him off quick.
    
7. In Dante Must Die mode, Vergil acquires a new move that DMC1 players will find very familiar – Summoned Swords. Stay far away from him when the swords still surround him, then roll/jump/dash away when Vergil shoots them. He has 3 different ways of shooting them, but you dodge them all pretty much the same way. Just be extra careful with the direction you're evading towards if he does Blistering Swords (his swords rapidly shoot at you one by one from his side). Vergil stays protected until all the swords are gone, so attacking him in that time will not be a good idea.

Old video demonstration (1.53 MB)