The Night King was headed for the wall before he acquired Viserion. How did he plan getting over a 700 ft tall wall made of ice, fortified with magic, made specifically to keep him out? Has he been moving toward the wall since we first met him? If not, did the hatching of the dragons signal him.
He's the result of the COTF magic. I understand he's tied to the children, the weirwood trees and he has some sort of greenseeing powers. None of that is related to dragons, plus the dragon were born in Essos. No weirwood trees in Essos.