You can purchase new cars from the Pit lane in the main menu.
When you have 5 owned cars, apply a different paint colour to them. Each respray will cost $5,000 so this achievement costs $25,000 just for the spraying. Including the 5 cheapest cars, you are looking at around $175,000 total cost for this.
Also, you must maintain ownership of all your sprayed cars until you have 5 in your garage that are sprayed.
You can NOT buy three cars, spray them, sell them, buy two more, spray them, and have the achievement unlock, for example. You need 5 resprayed cars in your garage at the same time.
Note that some races have a requirement to have the car you enter be resprayed (Tier 1 event list, for example). So some of these will come with natural game play.