I had a notion that a campaign could be set in the actual Blood War.  In my head, there are the usual horrors and dice-rolling and stuff, but I was thinking a lot of the flavor would be inspired less by Mad Max and more by Blackadder Goes Forth, or Catch-22 or even a really nihilistic version of M*A*S*H.  The idea is that the players are at the very bottom of the rigidly hierarchical army of Hell and trying to outsmart their own superiors to avoid being sent to their horrible doom.  There are skirmishes with demons of course, but those are survivable.  The problem is that everyone knows the day will come when the whole army gets ordered to go Over The Top.  And the knowledge that when that day comes, the order will probably not even be given with any hope of winning the war (which everyone knows is cosmically impossible), but rather as part of some Duke's scheme to embarrass another Duke.  And the inability to even express this dread because the chain devil political officers are always on the lookout for subversives to make an example of.

I'm not sure who the players would be.  I think maybe just lost mortal souls.  If that's the case, do they gain levels?  Can they learn magic?  Is the magic tempered with certain rules (ie, you're in Hell because you were a bad person, you don't get to learn Plane Shift and get out)?  

But maybe they're low CR devils?  I think the whole idea is a little depressing for my regular gamers, but I don't want to let a good idea go to waste just because I can't use it.

I don't know.  Any thoughts on how to flesh this out are welcome.

This could be really cool, what I would do first is round up any potential players and ask for a name and a backstory (why and how they got sent to the Nine Hells). Based on their past actions assign them a devil that lines up with their evil deeds. Give them that stat block to play as while in hell. Now obviously the leveling up is going to be the trickiest. Because I am lazy I would scour the internet for devil themed homebrew classes. If you can't find any then you'd probably have to make some yourself (which imo is daunting). If that doesn't sound too good then simply start them at "level 0" if you will (they'd still have the stat block of whichever devil they are). Do milestone XP, once the players hit the first milestone let them pick a class and add those class traits to the devil stat block. Rinse and repeat for level ups. Maybe for story you can create a faction of insurrectionists on the devil's side who are simply done with the war, it's been going on for eons after all. Their end goal could simply be to escape or perhaps even be absolved. Either way absolutely love the idea and good luck!

If your players are up for it you could have their characters be adventurers that all fell in the River Styx, lost all their memories (perhaps their levels as well), and are stuck in the Blood War trying to figure out who they are. You could even have made some character sheets that don't say anything on them except for their abilities, doesn't list their race or class anywhere, and they and the rest of the party have to figure out who they are and what they can do.

The Infernal Hierarchy is listed in the monster manual. You could bring them one step up each time they get enough xp.