Thanks to Uncaged Stalker for the absolutely best, easiest strategy to complete this. I am now going to improve and expand upon his method.
There are five races, each with six cars that must be used. Each race, those cars will vary.
Easiest method, if you have 4-Star driving skill, is to completely ignore these cars (skip to the end for the non-4-star method).
Rather, start the race with the first car in each map, then immediately drive back into the Agency Tower. Go to the garage and switch out to the Agency Super Car. Drive back and find the beginning of the race again. Park the Supercar a little ahead of the starting line, then hit select and restart the race. Now your Supercar will be right there, ready for you to swap out into!
This car is faster than all of the others in the game, so you'll have an easy time racing with it. When the race begins, switch into the Supercar, then wait for the other racers to finish their first lap (or you can backtrack to a good point and wait).
When the lead racer is about to pass you, start driving. Let them pass you one-by-one, and one-by-one, blast 'em to death and take 'em out. It's MUCH easier to defeat them this way than it is to try racing them.
Once you've eliminated all your rivals, complete the lap you're on and switch back to the original car you were driving. This way, you can save your Supercar for the other five races and hopefully save yourself the trouble of having to replace it - if it takes too much damage, it will begin to slow down and you'll have to get a new one.
The Agency Racecar is a tough one, though, when you have to go against it. I didn't eliminate/overtake the final driver until near the end of the last lap on the Pacific Tour race. Those Racecars have lower top speeds, but much higher rates of acceleration, so they're tricky. If you have too much trouble eliminating them during the race, you can (1) race legitimately, or (2) try the non-4-star method, which I will now outline.
This is the most common method of completing this achievement. Start the race, then backtrack and build a large barrier to block all of the competitors from being able to make it through. You can do this really well with the barriers from the track corners.
When the cars get caught in your blockade, destroy their cars - either by beating the crap out of them with your fists, or shooting them in the gas tank (I haven't tried it but I'm told if you lock on, then press the analog toward the back of the car, it'll lock onto the tank). If you're doing this method because you had trouble eliminating the others mid-race, using the Supercar's machine gun to destroy them at the blockade is a marvelous and easy technique.
Additionally, don't worry about destroying all of them on their first pass. There are three laps - you have three attempts to destroy each car. If a couple slip through, just build a better blockade and try again.
Lastly, once you've destroyed all of the rival cars, unblock the track, return to your car, and complete the race with ease. Good job, Agent.