The first race you are given Nitrous is the street for Part 2 of acquiring the Supra.
Hold
to use Nitrous and once the blue bar on your rev counter is depleted the achievement is yours. As this is a street race you can easily tell when the end of the first half of the race is as it shows the percent the race is complete, so use the full tank before 50%.
If needs be you can restart the race once you've done this.
Note: Nitrous only fires when you are accelerating and you can ONLY unlock the achievement in a race and not a showcase or rivals eventHere's a video (I use a manual transmission)