Hmm...
Well, this is mine; feel free to comment your thoughts
Using their childhood through a prologue to show how they came to being like they where in DmC (such as showing Dante's time at orphanages, and Vergil's time with his family) and then having a little scene to show the aftermath of their battle, and a shortened version of what happened in Vergil's Downfall, probably the most important parts.
Then you begin with the proper story, where we see how Dante and Kat have moved on with their lives (if you want to have a nod to the classic fans make it they've opened a business called Devil May Cry, not what I'd do in the movie though).
Dante has been travelling around the world finishing off demons, saving places like that, but he's sort of like this person who does it from behind the scenes, not letting anyone know he is there (but we don't know this until later)
Kat is doing her thing back in the in the city where all of the game went down, when she is approached by someone, telling her that Dante died. Kat goes off searching for him to see if it really is true, and goes to a place where this stranger said Dante died. She gets there, and again we see the stranger (yes it's a bit obvious who it is, I know) who then kidnaps her.
We now go over to Dante who is finding more and more demons wherever he goes, and they've been getting stronger too. Late one night, while out fighting, Dante is approached by this same stranger. Who tells him the same news he told Kat, and gives him the same the same place to go to. Dante suspecting something is wrong still heads there anyway, because he is concerned about Kat.
Dante finally reaches there, and it's been turned into someplace hellish, with more demons, and they're stronger too. Anyway, sequence of battles, more battles and they lead to them figuring out, it's actually Vergil who set this up, who wanted to kill Dante so no-one could oppose him... shock horror D:
Leading to him in a final battle with Vergil, which will decide the fate of the world... and Kat. Big destructive battle between the two, but stuff goes wrong, and Kat has to sacrifice herself to save Dante and the world, and now Dante has to go on by himself now.