Vergil was already lost prior to downfall...it's just that the events of downfall brought out what really was inside Vergil.
Before the DLC takes place he's just had his only family turn on him and try to kill him. Vergil is still shocked and betrayed from that, and the doppelganger uses it to his advantage in his attempt to take control of Vergil.
Just like Dante is taken over by a demon side, Vergil has one too, his doppelganger. And because Vergil is weak, his doppelganger is able to separate itself from Vergil. This is when the doppelganger starts to manipulate Vergil by taking his amulet, the source of his power, and begins to prey on what he knows are Vergil's weaknesses: His feels for Kat, Vergil feeling inferior compared to Dante, and how Vergil thought Eva loved Dante more.
It seems like wherever Vergil is (real hell or hell that his mind has made) that the doppelganger sets up these encounters with dark versions of Kat and Dante to make Vergil face them, kill them, and thus kill his ties to his last bits of sanity. These entities of Kat and Dante are exaggerated versions which prey on Vergil's fears and insecurities in order to make Vergil confront his own feelings about them and overcome them in his own twisted way.
The doppelganger does this because it wants to take over Vergil completely like Dante's demon side nearly did. But in the end, Vergil's need for power is so strong, made even stronger by his victories over Kat, Dante and Eva, that he's able to overcome the doppelganger and absorb it back into himself, thus having full control of his demonic powers, but at the cost of his sanity.
So, Vergil went from man with an inferiority complex and need to control, into a cold hearted killer who fully embraces his demonic side by the end of downfall, due to manipluation by his doppelganger when it set up encounters with twisted versions of people Vergil once had ties to, in order to make him literally face his demons about himself and his past.
Well....that's how I see it anyway....