The concept is inspired by the famous trollfic My Immortal, where the main protagonists were all "goffik". "Goffik" people wore almost exclusively black clothes, frequently slit their wrists, attended My Chemical Romance concerts, were Satanist and used vampire puns such as "fangz".
Avril Lavigne and Gerard Way are, causally, the first goth emos ever. Janet Roberts once mentions that Avril (who should not be mistaken for Roxa Lavigne) was living with her own music, even back when she was a teenager.
GLaDOS is the first goth emo seen in MarrissaTheWriter fanfics, appearing as such from the very first chapter of ITS MY LIFE!. She has become a goth emo because she was reminded that she used to be Caroline, despite her hatred for humans. She also successfully sewisides, but doesn't stay dead for long until Marrissa Roberts misuses her powers.
Marrissa herself becomes a goth emo when she is heartbroken by Wheatly, who Marrissa saw enjoying Chell's booty quake. Once GLaDOS reveals that it was her plan all along, Marrissa stops being a goth emo.
Marrissa becomes a goth emo once again in the non-canon fanfic MarissaxChell, once again, seeing Wheatly cheating on her for Fact Core. This time, her goth-emoness is a lot more inspired by My Immortal, and she listens to Texas is the Reason but not "Averill Levine", seeing as "only poser preps did that shit!".
After Marrissa, there is a long pause until Post-SCrash Session 3: Spectators of the Host, where Skepness Man Beauregarde has become a goth emo, due to being heavily restricted by his mother. He doesn't show many traits common to goth emos, however, as he doesn't listen to Avril Lavigne music or attempt sewiside.