"Don't bring up the fly thing, fellas. It really bugs him." - Raphael, punning at Barney's expense.
Barney Stockman is the twin brother of perennial Teenage Mutant Ninja Turtles mad scientist villain Baxter Stockman. People frequently mistake him for Baxter, a source of great irritation for him.
His only appearance is in Raphael Knocks 'Em Dead, which in the beginning while the Turtles were channel-surfing, Barney was seen mixing chemicals, suggesting he has his own science show. Barney and Pinky McFingers kidnapped Raphael and used him to spread chaos until the turtles rescued Raphael and threw them in jail.
His being Baxter's twin brother may have been driven by the animation studio wanting to re-use Baxter's basic model, unused since Baxter was mutated into a Human Fly. It seems Barney must have made some contact with his brother since he first encountered the Turtles, and possibly after he was mutated. Upon meeting Raphael, who thought he was a restored Baxter, Barney mentioned knowing about the Turtles and his brother's fights with them.