DiRT Showdown

Start by going to joyride on the main menu. Then select Battersea. You'll start of in an open area, and will be able to explore as you please. There are 75 missions you need to complete, over 3 parts of the compound, with 25 in each area. The later areas will be unlocked as you complete missions from the first area(s), and you can press X to bring up a list, and there is the option to mark an event on your map.
To complete the missions, you are tasked with completing various activities, including donuts, drifts and jumps. For the jumps, I advise taking a long run up and using boost before you leave the ground. With the donuts, you need to press B to start a drift around the designated object, but keep leaning your left analogue stick towards the object and hold down the accelerator. For the drifts, just press B whilst turning to the left or right and your car will begin to drift.
At 100% completion of these activities you'll get this achievement. At 50% of the way there you'll get the achievement for completing 50% of these missions.
If you complete all of the Yokohama compound missions as well, you'll also earn the Hoonigan achievement, for completing all missions in both locations.

Here are some really helpful videos, one for each area. Credit to MotoGamesTV