Haha, I used to do that with my D&D characters! I've also heard Squirrel Girl is doing some really fun stuff with her Twitter tying into the comics!
But no, for my comic I don't. I actually kinda try to limit how much contact readers have with the characters outside of the comic -- I want the comic itself to be where you meet and get to know them, so for my story, being able to follow them on Twitter would cheapen some things a little, since it's easier to see what they're thinking in a way.
Plus my characters are in a pseudo-medieval world, so anything they tweet would have to be non-canon extras by definition. xD I think it might be fun to have a twitter account for a character who could actually have one in-story!