If you want to get to the DLC without beating Way of the STRONG, go back to the World Map, hit Square to bring up the area select screen, hit square again to change the difficultly to "Way of the SAMURAI", select the "Tohoku Region" to start the Dragon of the North DLC.
How To Unlock Way of the DEMON(super hard) in Nioh
Beat The Game Twice
Yes, you read that right. Before you even think about punishing William further, you will need to complete two playthroughs of the game.
Way of the Demon essentially acts as Nioh's NG++. After beating the game for the first time, you will unlock Way of the Strong. Once you return to the game overview screen, you can choose any available mission on this difficulty. If you've got the DLC, complete all main missions on Way of the Strong and then return to the overview screen. Now you can get started on Way of the Demon!