I'd like to recommend doing the welcome to Hollywood race over and over again in arcade mode. Also by using the different vehicle classes (Bike, Tuner, Exotic, Muscle, Luxury) it will work towards the Champs. Please only use this method after you have finished beating the city champ, no need to grind really until after he's defeated, unless you just really want the achievement very badly.
One more thing, go for the True Dedication achievement before grinding this, because those goal attack races count towards finishing the 300 offline races.