I don't think he'll become king of the realm, I think he will be freed from his vow and eventually be named Lord of Winterfell (Robb's letter is somewhere out there, giving him the claim over Bran or Rickon right?) and King in the North.
Where his lineage will come into play is after Dany conquers Westeros with dem dragons, the information that Jon is actually Rhaegar's son will be enough for Dany to allow him rule the north amicably. (That, and the whole "we defeated the white walkers for you" card he'll be able to play... and that Selmy has always spoken highly of the Starks when Dany lumps them in with the rest of the "usurpers dogs" as she calls them.)
All of this would require Howland Reed showing up, with possession of Robb's letter, I suppose... He's the only living person remaining who knows what happened at the Tower of Joy, for one, and for two, the forces Robb deployed with the letter were on their way to meet up with him in the Neck.