The books are more explicit in detailing how the countryside has been devastated by the many wars (one factor leading to the formation of the Brotherhood without Banners), so with all the refugees coming in King's Landing has the biggest concentration of population in the south. And establishing a power base in the south allows the night King to attack any remainder in between on two fronts, making short business of the entire affair.Is there any reason why the Night King would care about the Iron Throne? Or King's Landing - other than a source of new dead people. There's a lot of potential dead people to recruit into his army between Winterfell and King's Landing.
... all of this presuming there's no mystical relevance to it that we don't know about yet. (Or the possibility of raising more dragons, if that works out.) And this is the main issue: We know nothing about the Night King's ambitions and personality – either there's something we don't know, or it's just "bring eternal night for the lolz" as Bran has stated, in which case there's no rationale to be had anyway.