I thought they had just a sibling rivalry thing at the start of the game, Dante says he doesn't care either way what Virgil is doing, he just wants to interfere because it's him, not because of the consequences. But then later in the game, he says his 'soul' wants to stop Virgil, even it means killing him. So his feelings about the consequences changed at that point. I don't remember him saying he hated Virgil though, and later he tries to save him, and also is sad that he has gone.