Going by the name Doom:
Once, Crash was Doom's military instructor.
Long after everyone else on Phobos was dead (the only other survivor came to be known as Phobos (character)), he kept fighting, determined to send out a warning to Earth before the demons could claim him.
Distracted and intelligent sounding, (but not a member of the intelligentsia, more like a well-educated tradesman) this guy may be a few cards short of a full deck, but his training serves him well. He's a bad-ass in the arenas.
In the chat lines he also get mentioned as "Mad man", "Loony" and "Schitzo". When he's losing the match, you can witness him losing levels of his sanity. Talking about voices from the pitch, massacres in the corridors of UAC and dead friends. You'll see that he's a reckless war veteran, even if he's called a lunatic, don't turn your back to him or he'll make you pay for your foolishness. He also mentions having gone into 'hell once' which seems to suggest he's based on classic Doom Marine between Doom 1 and Doom 2.
Doom devours Railgun and Rocket Launcher first, then tries to grab Quad Damage power-up whenever possible. He will also look for the BFG10K.
In Quake Champions "Doom" is established to be the Doom Slayer, and the uniform appears as one of Doom Slayer's alternate armor sets. The armor lore appears to set it apart as being another point in Doom Slayer's (originally Doom Marine's) history, and another mission to Phobos than occurred in Doom 1. The original Doom 1, 2, Final Doom, and Doom 64 armor appears as a seperate armor set.
Chronologically speaking Champions is said to be a prequel to Quake Arena games.
Behind the scenes
Like all the AI in Quake Games, its possible to have a chat with the characters, in which they have various things to say, sometimes funny sometimes related to their histories, etc.