Aaron is not intended to be LT's son. IIRC his back story was that he was more or less a troubled kid that wound up dumped in the Army. He had a distinct affinity for nature skills and tracking. Something that he shared in common with his civilian trainer LT. It's not well laid out in the story, but he looks up to L.T. like a father.
After Kosovo, when he gets back and starts having PTSD induced flashbacks he begins attempting to contact LT for guidance; since LT was his spec ops trainer it's a natural choice. LT doesn't have any experience with killing people (there's a kind of important scene in FBI headquarters that establishes this) and has no answers to provide. Being a warrior and rather stoic individual, rather than saying "I don't know" he just ignores Aaron to let him figure it out on his own.
Aaron can't move on, so he snaps, goes off the radar and starts hiding out in the forest. Then the government sends in two spec ops sweepers to eliminate Aaron since he's a liability. Queue rest of story.
The whole "kill me a son" thing is somewhat metaphorical, but it's also an issue with the director William Friedkin has a penchant for putting references to the bible in his movies. Which fits since he directed The Exorcist. But it kind of overpowers and dilutes a lot of the subtleties of this story. He also doesn't spend nearly enough time explaining the backstory and relationship of LT and Aaron, instead deciding to focus on the action.
The one point that I will give him is that in order to help create the story he called in one of (if not the) top trackers in the world as a consultant. The story itself, is supposed to be semi-biographical (though very heavily fictionalized) based on one of the tracker's (Tom Brown Jr.) real cases.