How long would it take to speed straight through the entire DMC series?

The games are not long at all, but they are way more difficult than your average game. If you're good at hack and slash games, you'll probably have no problem.

To just simply beat the games, and not worry about doing all the little secret missions or trying to get high rankings the mission (where much of the replayability comes from) will not be too taxing. You could probably knock each one of them out over a weekend.

Play 1, 3, & 4.

Absolutely skip 2, it's festering garbage.

The reboot/remake (DmC) you can also skip if you want, as it's a spinoff game not developed by Capcom and has nothing to do with the other games, and not as good as the mainline series (although it's not nearly as terrible as 2).

Like a week maybe?

I would suggest playing 1, skipping 2, playing 3, and skipping 4. You can play the reboot (DMC) if you want.

For level of difficulty easiest to hardest, it's probably.

DMC
2
1
4
3

the original PS2 version of 3 has a messed up difficulty level that I believe the remasters resolve that but it's still tougher than your average game.