This can seem fairly intimidating when you first get a derelict, but it becomes more grind than anything.
The first step is to find a derelict car. You unlock the ability to find certain ones after quest chains. Find the chassis and the parts, and you'll unlock:
Refer to the solution written by NE0182 in the 'One Man's Trash' achievement that has all the videos with timestamps on where to find all the parts.
Next step is to go to your garage and choose what type of car you want it to be. Choose carefully, as you can't switch after. Whatever choice you make is going to set certain defaults for the car, and lock it to only doing those specific races. I personally chose a runner with the Ford Mustang because I needed a good Runner anyway.
Now for the grind. The only real way to go about this is picking up the tune up cards in the shop or in races. Your derelict is going to start with stock 1's in everything. Use trade in cards to get roulette style parts you need, or just buy them when the shop restocks. The best way, however, is the speed cards at the end of races. Your car has a performance level (starting at 100) and the level of cards you get is directly related to the cards you have and that level. Currently, there's no way to use another car for higher level cards to use on the weaker car. You just have to work your way up. So if you all 1's, the shop will stock 1-3 range, and you'll be rewarded that range at the end of races no matter what race you do.
As you start getting higher level cards, you'll start getting higher level rewards, and you'll eventually work your way all the way up. Since this achievement requires level 399 anyway, just keep this car and use it on all the associated races in the story to keep upgrading it. You'll get the achievement when all of your cards are level 18.
A great race I found for the runner car to grind is the Carville Trial, unlocked during the runner questline in Chapter 4. This race is found in the lower right of the map, near two drift events and another runner event. This race is just a Point A to Point B race that guides you on a rather short and simple path. There is a gas station a little south and west from the start point that's very close to fast travel back to. The time is generous, and there isn't really anywhere to really mess up. The side bets are rather easy as well, including 'Avoid all cars', 'Destroy X objects' and 'Drive clean for X seconds', for example. The side bets give a lot of money, about enough to where you can afford an additional card from the shop and get the free card from the race. If the card you get in the race isn't good enough, trade it in.
Choosing drag can be easier if you've mastered shifting. Any of the races that feature best two or best three will work. Start at the lower level ones and work your way up to more profitable races when your car is strong enough. Most of the side bets require wheelies. If your car has a high enough performance, it will do it automatically, and you'll know it when you see it.
You'll also get two other achievements from grinding the race.
If anyone has suggestions for races to grind for the other classes of cars, leave a comment and I'll add it to the solution.