Of course he knows. If Trish put it together not knowing Urizen was Vergil then Dante probably did so faster, too, knowing it was. He goes around with demons named Griffon, Shadow and Nightmare and if those aren't enough hints he calls himself 'V,' carries around Vergil's favorite childhood book of poems, and it even has an active knowledge of him and his history. There is too much of Vergil in him and a lot of it is things only Dante would know and if he didn't know at first he figured it out before long.
Now, did V lie to Dante? Absolutely. V's intention was never to end his life along with Urizen, it was to reunite with him to be whole again but he probably lied to Dante about it to gain favor. 'Now that I'm all human I know what I've done.'
If you look at the way they interact with each other it makes sense how they talk to one another if he knew rather than not. Play the game again and with every line and cutscene assume that Dante does know, from the start, who V is, and every line and look takes a different meaning. Some of it is all vague, sure, but only because the game wanted to keep the whole thing quiet for a surprise but from what I saw, knowing that V is Vergil, Dante definitely gave off vives that he was. There is even a hint of animosity towards V at times, as though he is getting judgemental.