*heavy metal music played behind a Ken Burns-style pan over footage of a family tree*
So in the 1940s, William Joseph Blazkowicz fought Nazis. After the war, he came home, got married, and had a son, Arthur Kenneth Blazkowicz. When he got a job in television, Arthur took the stage name "Arthur Blaze," a name which carried over to his own son, William Joeseph Blazcowicz II, aka "Billy Blaze." Billy apparently dropped the affectation when he grew up, which is why his son can be William Joeseph Blazkowicz III. BJ3 (as he's probably not known) joined the Marines, savagely beat his commanding officer for unlawfully ordering him to fire on civilians, and was transferred to a base on Mars where he became solely responsible for repelling a demon invasion from Hell.
(There are several different continuities now, but Commander Keen remains BJ Blazkowicz's grandson. And according to creators John Romero and Tom Hall, the original idea was that ultra-heroism in the Blazkowicz lineage would skip a generation, meaning Doomguy would have been Keen's grandson, not his son, but still: it's a big, happy, anti-fascist family.)
This Doomguy is based on the modern games, not the old ones - maybe Funko was hoping to do a retro set later, but the Best of Bethesda line came out in 2016 and Funko barely seems to make Mystery Minis at all any more, so that never happened. The armor is chunky and lumpy, as it should be, and he's armed with a Plasma Rifle. The helmet and boots are a different shade of green than the central body, more olive than teal. There are grey sections between the green plates, a few red accent spots, and a silver frame around the dark green visor.
The Blazcowicz clan probably continued at one point. Until it was patched out, a torn photo in Doom Eternal showed a woman holding an infant, with the implication being that was Doomguy's wife and son, but little William J. Blazcowicz IV was killed when the demons invaded Earth. That loss (and that of his pet bunny) is why Doomguy is so unkillably angry.