Mega Man 10

Probably one of the easiest stages to do is Wily castle 5. If you want to try to get it on an earlier stage, try going on Pump Man stage.

Thanks to KMetalmind for this tip: Another good stage is the Commando Man one. If you take the upper path, it should be easy to finish the game without getting hit. The only enemies who usually can make you lose are the Mets (enemies that hide in their hat and shoot at you). If you are careful with the bombs and with the bugs, it´s a piece of cake :)

Just to add to Shi Kiseichu's solution, obviously this is much easier on, well, easy mode. I personally got this on sheep man stage on easy mode.

Start a game on Easy mode and play through Commando Man's stage. Proto Man is good when taking the lower path for sliding under the red springing enemy. Just reach the first boss door and the achievement will unlock when you enter the empty room before the boss.

For those who bought DLC, use the character Bass to battle through the stage along the top path. Shoot up to take out the green flying enemies before they shoot mines. Shoot the enemies with helmets that bury themselves in the ground at point-blank range holding down Bass' gun even while it's buried, so that it pops up to run into your gunfire. Bass makes short work of the level and this is the easiest method for earning this achievement.