The first chance you'll have at this achievement is in Chapter 3, when you do a drag racing questline. You'll see a meter at the top of your screen that has a yellow zone, a green zone, and a red zone.
As you race, you'll see a line move up the bar. When it reaches the green zone, press
to shift up. It will then say 'Perfect shift' if you hit the mark. Do this each time you shift during the race, and at the end of any round you'll unlock the achievement. The rounds are rather short, and the hardest part is when you're starting the race and the line is moving quickly through first and second gear. Most of the tracks are straightaways, so you can focus more on the bar up top instead of avidly staring at the streets.