Carter Matheson is an agent with the Bureau.
This is definitely, absolutely not the FBI. That fine establishment's role is to investigate federal crimes and arrest the perpetrators. The Bureau for which Matheson works directs him and a handful of other operatives like him to kill enemies of the state.
That is his main purpose. As he tells us succinctly at the beginning of the first recorded adventure, "My name is Carter. I am a hunter. I hunt bad guys". What he does not say explicitly is that he kills them when his hunt is successful. He does not have to, really, because to him, it is implicit in the term "hunt".
Matheson lets us know in a prologue of that first tale he was always very good at hunting. He got his first bow and arrow set when he was eight and learned how to use it. Then came a BB rifle at around nine. Then knives around ten. Then a .22 at age 11. And so on. But while the power of the weaponry increased as his age did, he kept up with the previous items. By the time he was old enough for the military, he was already an excellent shot.
His father was an officer in the Marines, a tough old bird who lived and breathed the Corps and brought his son up knowing that being a jarhead was the way to go. Young Matheson joined the Army. I think that says a whole lot about the relationship.
He excelled in the Army. Went on to become a member of their Special Forces. Honed his hunting skills to become one of the best. Got noticed by the ultra secret Bureau. And now works for them, doing what he does so very well - hunting.