Hotshot Racing

This can be done in local multiplayer
When choosing your game mode, before selecting it with A, press over on the D-Pad to make it two player. Any track, any number of laps MORE THAN ONE (comments have confirmed you can not earn this in a one lap race). Just set the number of AI to zero.
Now all you have to do is drive carefully enough to avoid hitting walls, while maintaining speed so you don't drop below the threshold and take damage. You don't have to worry about crashing into another car.

Castle Funfair
Single Race
Game Mode: Drive or Explode
Difficulty: Normal
Laps: 3

Character/car: just use whatever you're comfortable with.

It's pretty simple once you get a hang of controlling your throttle. What you want to do, is to press RT (acceleration) lightly during rolling start, so you're just bit over the minimum speed. It's important, because you want to stay behind other cars, to avoid them crashing into you. Once you're getting near the checkpoints, remember to add little bit of power, as the speed limit will increase.

Remember, if you drop below minimum speed or collide with other vehicle, you will lose health which will void the achievement.
As long as you stick in the back and be careful not to drop below required speed, you will get it in no time - fortunately it's only one lap and not the whole race... :)

Start a Drive or Explode single race and keep the difficulty on Normal. Stay toward the back of the pack, but keep your speed up so that you don't take damage from going below the speed limit. Keep your space from any other drivers and the walls and you should easily be able to complete a lap without taking any damage.