this is pretty easy to do, just select a car, i chose the 1st derby car as it's cheapest, then carry on racing and when you have enough cash go to the garage and then upgrade all the car parts to MAX, then go to the 1st derby race in the career which is pretty easy as the AI are not that fast and just win the race and the achievement will unlock.
you don't have to wait to upgrade the car, just upgrade it bit by bit and keep racing until you have enough money to fully upgrade the car.