I feel like I should say that the animosity between Dante and Vergil was explained in the DMC3 manga, which I'm pretty sure is canon.
Here's an excerpt from the DMC wiki summary:
"Vergil reveals his plan of raising the Temen-ni-gru, thus opening the gate of the demon world. When Dante declares his intent on stopping Vergil, Vergil then proceeds to taunt Dante by showing him the half of the amulet that Alice stole earlier, prompting him to attack. However, after a quick battle, in which Vergil trounces Dante, Vergil throws his brother’s half back to him; Vergil then taunts Dante again, stating that he could retrieve the amulet from him whenever he wanted."
A year later the events of the DMC3 video game happen.
You can read the manga yourself on mangareader.net.