In order to obtain this trophy, you will need the Nitro 3 upgrade for your bike found in the Diamond Lake Camp, since it's the only one that lasts for more than 5 seconds. It only becomes available for purchase once you reach Trust Level 3 (See  Best Friends Forever (For Life) for more information). Do all story missions, side quests and so on to get as much Trust Points as possible. The best weapons to fight Hordes are the "Chicago Chopper" and the "RPD" machine guns (found in Wizard Island, avaliable for purchase on Trust Level 2), so clear a few in Highway 97 (they get marked on the map after the final mission) and eventually you'll get the 25,000 Trust Points required for the Trust Level 3 upgrades.When you get the Nitro 3, go to a flat, open area with your bike and press  or  +  +  + . You have to accelerate a little and drive in a circle (making "donuts" doesn't count). It might take some practice, but should unlock after a few tries.

Burnout Apocalypse Video Guide

Credit to PowerPyx for the video.