I can confirm that each of the three runs needs to be flawless. If you get hit by a bomb at any point you need to quit out and restart the level.
Edit - TwinTalisman
says that killing yourself (by jumping into the water) restarts the whole challenge and resets this achievement. So no need to quit out and do it all over.
Start by clearing out every enemy in the level so the path is safe. This'll also give you enough spirits to unlock the super flame powerup in the gazebo, next to the end portal. Talk to the character next to the portal who will activate the challenge.
It consists of three stages where you have to activate the super flame and shoot at Bombo, who'll be holding the flags you raised through the level. The first time he's at the last flag you raised, the second time the second one, and the last time he'll be at the first flag you raised at the start of the level. He'll come down, talk to you, and then start a sequence where you have to follow him to the gazebo while he throws bombs at you.
The safe method is to take it slowly. There's no time limit, and he never moves backwards. Get close enough for him to throw a bomb and move back while he moves forwards each time. When you get to a certain point he'll throw the flag into the gazebo and move to the next flag.
Here's a clip of me completing the last stage of the challenge.