He was portrayed by Marton Csokas, who also portrayed Mr. Smith.
Teddy's biography is extremely unknown except that he is a skilled hitman.
Going out to Kill McCall and Death
Teddy is sent by his employer, Russian mafia boss Vladimir Pushkin, to travel to Boston and kill Robert McCall (the protagonist). He spends the entire film pursuing McCall and attempting to capture him. It is revealed that Teddy's real name is Nikolai, and that he used to be a Spetsnaz. Later on, Teddy and his men break into Home Mart (where McCall works) and hold everyone hostage and threaten to kill them unless McCall surrenders. Having already killed a number of Teddy's henchmen, McCall creates a chemical explosion that kills the last of Teddy's men, and kills Teddy himself with a nail gun.
Plain and simple, Teddy is a violent psychopath who takes pleasure in killing and torturing, evidenced when he beats a man into bloody pulp over 'nothing' and calmly strangles a girl to death for not telling him where McCall is.
And thanks to his military background, he is an expert at hand and gun combat.