Why was The Accountant feeding information to King?
King was The Accountant's "inside man" at the Treasury Department, and in a vague explanation via the film, it's revealed that The Accountant has a hypocritical moral code of decency and in this backwards utilitarian justification it makes sense to have someone on the inside who's running the investigations that could/would lead to The Accountant.
He let King live in NYC, and King has a dreading sense that at any moment he could be found and killed, and/or also that his family may be killed or harmed. It's not that The Accountant was necessarily feeding information to King, it's more like King was letting The Accountant go about his business and keeping him out of the records and official reports. He then recruits Medina, because he can blackmail her with her juvenile felon past, to be his replacement. Again, it's all a weirdly gray -- if not outright black -- moral utilitarianism that somehow justifies the ongoing existence of The Accountant, mostly out of fear of reprisal.
He didn't feed King information, he kept him in a place of influence so that he could call upon him when he needed to, because King was afraid of The Accountant.
It's slightly, if not highly, non-sensical ... much like most of the movie.
Director King tells Medina, in the scene where he explains what he knows about the Accountant's history, that he came to realize that the Accountant was turning in criminals who broke his moral code. What his moral code is was not explained. King would then create a "smoke and mirrors" investigation to cover up the true source of the information.