This achievement is pretty simple. You must win 5 races with a custom vehicle, this can be done a number of different ways but the way I did it is I went out and stole a car, which happened to be a "Compact" car. You can tell what kind of car it is because it will say it at the end of it's name. Then you need to take this car to Los Santos Customs or any car shop and put an engine upgrade on it and a tracker, I think you have the engine upgrade at Level 1. There may be cheaper mods that will classify your vehicle as custom, but I was playing safe. You also may not need the tracker, I'm not sure.
Now you must find a race that will allow you to use your class of car. There's a race at the prison near Sandy Shores that will allow you to use any class and take about 30 - 45 seconds a lap. Go to the job spot of this race and enter the circle (or go to the Job section of the online menu), it will take you to a menu, choose to host your own game. Set the rules to 1 lap and make sure custom vehicles are on and the class is set to the class of your vehicle. Then invite your friend, and be sure to pick your custom car (it should say "Custom _____") Then simply win 5 races, at the end of each race you can just hit "Replay" at the bottom of the next mission select and replay the same race.