For both Dante fights, just spam your Snatch. Having it upgraded to Level 2 is best. To keep him busy, shoot at him so he is too busy deflecting bullets. Walk towards him, then jump in the air and snatch him. Then slam him to the ground.
If you snatch him on the floor, chances are he will do a Stinger on you, so roll out of the way after you snatch unsuccessfully. If you try it again, he'll probably Stinger again. So for majority of the fight try to constantly jump towards him in the air, and just Snatch and Devil Buster him.
If you connect on the floor, usually a scene of them clashing swords happens. Several things can happen here, depending on the style he's using. Royal Guard, it usually ends in a stale mate, and you have to slice him quickly or he will slice you first. Gunslinger, he usually gets the upper hand and shoots you (which is why you buster him from above). Sword Master, you will deflect his word and fling him away, damaging him.
It is not a very hard fight. On the upper difficulties, he starts using Pandora a lot.