The easiest way to do this is on a challenge as you have unlimited boosts, overtakes, directions and whatnot. Trains and helicopter are also free. I did it on a medium one (6min 30s) and got it easily. Just boost and overtake around the city.
I did all the three vehicles first, then collected the coins and finally finished the mission. Unlocked straight after finishing the challenge. BTW unlocked 5 night missions achievement at the same time too.