There are four cases to be solved over the course of the game. These are a good source of Cop XP, so remember to take the right clothes to them.
The first case is Popstar, and completing this case is mandatory to continue the story. Hotshot is similarly mandatory and involves quite a bit of racing. Notably, you get the High Roller Tuono out of this, which is a decent Class A car. Start using this regularly to build up progress towards Reckless Driver.
The cases from here get decidedly less action-oriented until the end of their respective cases. The Serial Killer case starts after the story mission Loose Ends, but can't be completed until finishing Final Kill. This case involves a lot of disguises and fast talks. For the last case, chase down the suspect, action hijack the vehicle, then drop it off at the proper location.
After completing Fast Girls, Ilyana will call you to initiate the Kidnapper case. This one can be started and completed immediately. Track down the apartment, travel to the gambling den and intentionally lose to the suspect, then follow his boat, enter his hideout, and after a brief shootout, rescue the girls. The achievement pops after Teng's debriefing.