Funny, I've been wanting to talk about this concept for a while now.
While they have the same concept of "Green demon-slaying space marine," the details are all quite different. Just by looking at them, you'll see different aesthetics: Doomguy is rough and brutish, almost like a high-tech barbarian, while Master Chief's armor is clean, shiny, and cared for-Nothing less would be expected from a space soldier. They're different in terms of gameplay, too-Doom is fast-paced carnage, where you die if you stop moving for just a second, while Halo is slower and more tactical, with added emphasis on taking cover.
Doomguy would be a fast fighter who's surprisingly glassy for someone of his build. He would definitely hit hard, and his guns would be stuff that would compliment his fists in dealing damage, such as the high-power Super Shotgun and the rapid-fire Chaingun. For the sake of simplicity, he would have only one ammo meter, which fills up by dealing damage with his melee-based normals. He might also have an armor gauge as well, to help circumvent his glass cannon status. Like the ammo, it increases by dealing damage, and decreases the amount of knockback he takes. Every hit decreases the armor he has.
Master Chief, on the other hand, would be slower and tankier, with more emphasis on his guns. He'd have a fairly high jump like in his home series, and his run speed would be pretty sluggish, since the series didn't even have a sprint button until Halo 4. His gimmick would be that he uses his firearms in his neutral special, but only one at a time. Each gun has it's own ammo, and once that runs out, he switches to another gun. This is because unlike Doom, where you keep all your weapons, in Halo you can only carry two at once. Like Doomguy, he has armor, but it's static-It gives him a set amount of super armor and breaks after taking enough damage, then only comes back after waiting a while. His other specials would bring in grenades, barriers, and other Halo weapons.
Note that I haven't played Doom or Halo, so I may not have represented them properly.