Before starting a race you can add up to three Mods by selecting the "Mods" option which is found to the right of "Start Race". You need to add one of each Mod card (Green, Blue and Purple) to your Mod slots by pressing
on the card you want and
again to add it to an empty slot.
Some Blue Mods can only be used on specific tracks, where as others can be used on any, so make sure you have one selected that will work with the track you are on.
Green Mods are boosts, so select any you want.
Purple Dare Mods vary in difficulty. I've had some that start you in a specific position on the starting grid and another that makes you race with manual gears and clutch, so you'll need to make sure you have one that isn't too difficult for you skill level so you can still win the race.
Once you have one of each colour Mod equipped, start the race. If you finish in first place the achievement will unlock as you are being shown the rewards screens.