Personally, I think Reddington himself worked for the FBI at one time (and maybe still does), but he had to go on some balls-out mission that was so in-depth, so involved, etc., that he literally had to sever his ties with the FBI and "become" a criminal so that one day (now) he'd be able to bring the worst of the worst to justice. Somewhere along the line, he either got too mixed up in the whole thing or (maybe) he just straight up lost his mind when he found whatever it was that happened to his family, and in either case he ended up literally becoming the very thing he once sought out to destroy.
Some of the reasons I think this include:
- The obvious -- the very first sentence of the very first episode
- He knows their protocols too well
- He knows too much about too many people, many of whom are "important"
- Whoever Alan Alda's character works for appears to be connected with the FBI
- His connection with the person who was revealed to be the mole; also, see #1 above
- In episode 14, Madeline Pratt, Red appears to be basically admonished of any suspicion regarding Diane Fowler by an agent from a different FBI field office; aka it looks like he's being protected from the FBI by the FBI
As usual, I could go on and on but I'll stop now.