Make sure to select continuous speed run from the main menu, and once chapter 1 loads and you have control of Nathan, enable the donut drake skin. A few words of caution:
While a continuous speed run does not to be completed in one sitting, powering down your PS4 while you're in the middle of the chapter will remove the skin and disable it from being chosen again. You can safely turn your PS4 off at the start of a chapter; just remember when you boot the game back up to re-enable skin before you go anywhere.
You can safely use the rest/resume mode feature on the PS4 mid chapter, as long as you don't fully power down the PS4.
In addition, using any chapter skips will cause this not to pop as well. I have not tested any of the mid-chapter skips (like in 5 where you can wall shuffle through the gate and basically disable most enemies), however I did my first "continuous speed run" using chapter skips and neither this trophy nor the 2:30 trophy popped. The trophy will pop after the cut-scene, not once it starts like the other difficulty related ones.
Edit: As pointed out by bravezach in the comments - this can not be earned on explorer difficulty.