Edit:- All credit for this solution goes to Hirsuite Dave from True Achievements.
TrueAchievementsPretty self-explanatory and you'll probably get this on the first one you try.
While in career mode, you'll get chances to complete vehicle challenges - displayed on the far right of the career "desktop." All you need to do is enter and complete the course in the required car, under the required time.
Times are variable based on difficulty and can be changed during the event without penalty - though you will have to restart.