Early on, some fans complained that Emo Charlie actually fit the goth stereotype more than emo in terms of personality and style. (Source) Especially at the time of the character's creation, the stereotype for an emo guy would have been a soft-spoken wimpy guy, where as Emo Charlie was always confident and had an aggressive personality. As for style, the black lipstick is almost always associated with the gothic style and it was a bit unheard of for emo guys to wear lipstick at all.
Fans today don't seem to mind Emo Charlie's goth roots. The reason early fans felt compelled to complain about the details of the character was probably because when Greg first began using Charlie in 2008, "myspace emo" was still rampant on the internet. The "myspace emo" style and subculture is said to have died out in 2010.
Greg seems to have two personalities for Emo Charlie depending on how he is using him in a video. Sometimes he is calm, monotone and the voice of reason. Other times he is emotional and irrational. Greg usually uses this version of Emo Charlie when mocking self-harmers.