The easiest way is to just start near the back of grid (default position) let the other few car go past and then follow in 16th place. (you don't need to win the race).
I did this and the achievements below all in the same Free Play Race.
If you set it for a 4 round event, you can also get;
Underdog, Won an Event having finished the first round in last place.
Brawler, Beat your first Nemesis.
Show Off, Won a race by crossing the line backwards.
Coupon Car, Won an Event with at least 3 other racers using a loan car.
Check other posted solutions on how to get the above.