In case someone wants some actual gameplay tips instead of "just boost it", here goes.
First of all, you want to be playing as the Slayer to maximize your kill potential. Playing as the Demons is easier in my opinion, but you can only get one kill per round, if your Demon buddy doesn't steal it of course.
My strat was simple, open the round with the rocket launcher's lock-on burst, empty the entire gun into whichever demon is the slowest/has the least health. For example, Revenant may be hard to lock on to, but if all rockets connect, you can wear him down in one go, on the other hand Mancubus is hard to miss but he's got tons of health and your spent rockets won't earn you a kill. Furthermore, if you manage to kill one of them, they'll drop resources which will replenish about a half of your rocket ammo, which means you're sort of back in business. Nice!
Once you run out of rockets, I'd recommend two options - switch to either Super Shotgun or Chaingun. Super Shotgun is perfect for every type of demon, you can close some distance between you two with the Meathook and do massive damage with each shot, of course every miss hurts and it eats up ammo quite fast. Be extra careful when getting close to your enemies as you might get walled in by Cacodemons and Arachnotrons which is a death sentence, just quickly pop in and out.
Chaingun is great for the likes of Mancubus or Pain Elemental, the chunky slower guys. They won't require that much accuracy and a small dose of the Mobile Turret burns through their health quite quick. Just watch out of Pain Elemental's annoying shield or Mancubus' devastating flamethrower, keep your distance!
I personally wasn't using my chainsaw that much, almost not at all since it usually results in a loot block and enemy demons catching up to you. If you run out of ammo for all the guns I mentioned above, you can make a last stand with either the Ballista or the Plasma Gun. Plasma is basically like the Chaingun, good for the chonkers, bad for the quick guys. Ballista requires a bit more skill and accuracy, I would honestly just wait until my outline was hidden, charged my Destroyer Blade around the corner and waited for someone to show their face before I get murdered by the fodder demons.
Keep an eye on demons healing, they need to stand in the healing zone which makes them an easy target, especially during the 20s respawn timer for their buddy.
For upgrades between the rounds, I usually went for dash recharge, double equipment charge and resource drain. Picking BFG as the final upgrade is sort of useless for the player Demons (the tendrils don't damage them and scoring a direct hit is nearly impossible) but can be good for clearing out fodder, especially if two Barons appear.
Depending on how confident you feel about your opponents, you can just kite them around the arena, kill one, wear down the other before the first one respawns and repeat the process. Sometimes you "luck out" and the match will take forever even without you trying, even skilled opponents can be a blessing in this case. I scored 12 kills in one match just because one of the demons was really great at evading me while waiting for his buddy to rise from the grave. You won't get the same results every match, but don't get discouraged by 0 kill matches etc., it happens. I was able to pop this in around 4 days of casually playing the Battlemode, so it's not that bad. Just rip and tear until it is done. Happy hunting!