This is a story achievement, and pretty much can't be missed unless you choose not to do any story missions.
As with every story mission, you're tasked with 1-3 objectives that you have to complete in order to win the car. None of them are very hard and they pretty much all boil down to "Beat this guy in a race" over and over again.
Once you win the race and take the car back to the garage the achievement will unlock. I have no clue why it waits until the car is in the garage.
There's a trick to make the game go extremely quickly, however. Before launching the race, select the difficulty options and take a look at them. There are no difficulty requirements in the game, so if you want to speed through an easy completion, simply turn the difficulty down to Easy mode. It's virtually impossible to lose any race on Easy mode.
I would strongly recommend you turn off the assists, though. If you deviate too far off of the green guide on the track, they take over and it can get really annoying when you're trying to take shortcuts.