All you have to do is
press while standing near a police car. This will result in Lincoln opening the police car door, throwing out the police driver, and getting in the car. The achievement pops even if the dispossessed police driver turns around and kills Lincoln.
This is one of the simplest achievements in the game. It can be done at the very beginning of the game as soon as you have control of Lincoln and can move around New Bordeaux. The video below shows the achievement being obtained in the very early "A Taste of the Action" side mission where we are supposed to drive Ellis to Sammy's. So, interestingly, Ellis is with us for the achievement.
The below video demonstrates getting this achievement. To make it easier, we ram the police car boxing it between Lincoln's car and the car behind it. That way, Lincoln does not have to chase it:
If we stay in the car for two minutes being chased by the police, the following achievement will unlock after escaping the cordon:
At the end of the above video, you'll notice a green dot behind Lincoln on the mini-map. That's Ellis who has stayed with Lincoln through thick and thin.