The first key to earning this achievement is purchasing the Mitsubishi MiEV Evolution 2, a category 2 Hillclimb vehicle. It has tons of power and sticks to the road in all weather conditions with its 4WD transmission. The car costs about $120,000. You should immediately be winning one star, two star, and three star events with clean driving, even without upgrades.
The second component is actually getting the Hillclimb championship to become available. As Capt Poopey has noted in other guides, the game design is such that the fastest way to get to the event is using activity quitting. In my case, I was lucky to have it open up after quitting a two star event that had two races (some two star events only have one race, and didn't seem to open up higher events for me when I quit them). Choose the event, wait for the first race to load, and then drive over the start line and immediately pause and choose "Abandon Activity." Then, cross your fingers.
I had my Mitsubishi fully upgraded to 519, and won each of the three races of the Hillclimb championship by over a minute. I definitely could have done it with many fewer upgrades and saved some money.
A final note: Capt Poopey has also theorized that the difficulty of the event may be based on your particular overall crew skill level, with a higher skilled crew causing the AI opponents to have much faster times. My crew skill total was 76 when I earned this achievement, with most of my crew members under level 10