Quake 2 and 3.

I never played any console games online, and the only PC game I've played online is Blood. That was fun...when one guy killed the other the game would say things like "Reamed!" or "Sodomized!" or "Anal Vengeance/Anal Madness!" And when you accidentally killed yourself, it said "Scrotum Separation!" ...But that's more old school than what you mean, I think Blood might even predate Doom.