This is not a difficult achievement, but as there wasn't already a solution I thought I'd show you the route I used to earn it. At the end of the case I had caused practically zero damage costs, ensuring that I earned a 5 star rating.
First of all, when pursuing Vernon Mapes you do not need to follow him between the houses or down all the small alleyways. Doing so increases your likelihood of driving into a fleeing pedestrian. In fact, you can stick to a main road route the whole time which I show in the video.
Two Military Police cars will enter the chase towards the end. In the video I nudge the first one causing it to crash, but this is unnecessary. You can pretty much ignore them as they eventually crash themselves.
If you use the route I show in the video, there is no real time pressure, so take it easy. Keep a good distance so you know when to make the turns and make sure your sirens are turned on.
Note that you must do this "on your first attempt". So if you mess up (cause too much damage or spin out) and/or lose sight of him, then before you fail, go to the menu, select Quit, then go to Resume. Annoyingly you will find yourself outside the house before it gets torched by Mapes, so you will need to go inside, find all the clues, and escape, but it will allow you to bag the achievement.