Complete all the missions (on the GPS map they are the flashing ones with the exclamation points).
when you reach 50% completion make sure you have 35 races under your belt for each of the car types (Tuner, Muscle, Luxury, Exotic and also Sport Bikes) so that the Champs will appear (not including the Lowrider and SUV which have no champ, there is also no champ for the police vehicles).
When available, buy Karol's garage (cost, $1,000,000, not really that hard to get by that point in the game) and of course beat the City Champ when he challenges you.
nothing else counts. Goal Attacks, Arcade, Online, Collectibles(only unlock cheats), nothing else matters except the main missions.
Note: The Tournaments will reappear after you beat them on hard for the DUB cars, as will the 'zone' Matches (Hollywood Match, Beach Match, Downtown Match, valley match, etc.) and the Wager 'missions' that require you to win 3 wagers. These 'reappearances' of these missions do NOT count toward 100% completion.