To get the "Barrel-ly Made It" Achievement in Hotshot Racing, which is part of the free Big Boss Bundle DLC all you need to do is finish a Barrel Barrage race on fire.
To be on fire your health needs to drop below 25 out of a maximum 100
Hitting a barrel costs upto 20 health points, so just race as normal, drop barrels, and then on the last lap keep a close eye on your health, get it below 25 and make sure you finish the race and the achievement will drop.
You do not need to win the race.
Video here