I first noticed the phenomenon of TV characters twittering (er, actual people twittering as TV characters) when it caused a little ruckus during the last season of Mad Men. AMC initially protested the idea of fans twittering as Don Draper or Roger Sterling but eventually let the twittering continue — and even interviewed the fan who tweets as Peggy Olson after she won a Shorty Award.
In addition to the Mad Men crew, I follow Coach Eric Taylor from Friday Night Lights and Rachel Berry from Glee. I'm not sure who runs those Twitter feeds (my guess is that Rachel's is done by Fox, while Coach's is a fan), but regardless, it's fun to see people try to channel what the characters would say on Twitter — kind of like fan fiction 2.0. What about you?