Hold up while running through the screen that is before the screen where nidhogg consumes you.
It has to be the screen where the text "Final Screen" appears and the entire time you need to be running and have your sword raised.
Your best bet is to just run the whole level like this after your first kill.
If you die and respawn on the "Final Screen" you cannot earn this achievement without losing a screen to your opponent then killing them and moving back in to the "Final Screen".