There is no real basis for doubting the truthfulness of this historical account. It involves real people and real places.
Other than the story as it's given in the bible is internally inconsistent and so at best only contains a kernel of truth?
David is made an armour bearer to Saul because he can play music to chase off the evil spirit God send to torment Saul (I know right?). Saul knows him, and his father. An armour bearer would be in close proximity to the person who he is attending. Trusted to be in close proximity. David then turns up to fight Goliath where Saul doesn't recognise him whatsoever and hasn't a clue who his father is. After killing Goliath, Saul keeps him at his court and David works his way up the ranks etc etc.
If you think this part of the bible is a collected edition of tales about a semi-mythological founding father of a royal dynasty, this makes sense. David gets all the good stories attributed to him and they all get written down even if they don't mesh neatly into one biography. So, right from the start, you've a good indication there's darn fine reasons to doubt this account is either historical or truthful.