EDIT: With the introduction of the Race Shop, you can now just purchase mods, including legendary mods, making this achievement far easier. The mods refresh every 6 minutes so if you don't see any legendary mods just wait and they'll appear sooner or later.
All you have to do for this achievement is equip 3 at once and finish any race on any track in any car.
Legendary mods are obtained by purchasing Premier Mod Crates, although they are very rare, or from being a VIP, which gives you 5 to use.
EDIT: According to Xthunderspeeder having the VIP pass no longer grants you 5 legendary mods.
EDIT: Thanks to Spaceman001 for confirming that this also works in free play.