Going for the no damage achievements is actually very easy using an exploit:
Everytime you back out to the main menu you get an autosave. So keep that in mind.
After you've cleared one or more (how many you like) rooms without being damaged exit to the main menu to get another autosave.
In case you get hit you have to either let enemies, spikes or bonfires kill you or bomb yourself to death. This can be a tad annoying if you have 12 hearts towards the last couple of floors like the "Womb/Utero" or the Darkroom/Chest.
NOW THE IMPORTANT PART!!!
During the death animation pause the game and hit the guide button. Manually quit the game.
VERY IMPORTANT!!!
You MUSTN'T quit after being hit BEFORE you die. That way it would for whatever reason save your current health.
VERY IMPORTANT!!!
If you continue after starting the game up again you will be back where your last autosave was.
This works for THE LOST playthroughs as well!!!
Of course with him killing yourself is easier since he has no health and dies in one hit.
Items like Holy Mantle (you can take one hit every room), the "9 lives cat head" or guppy's collar ( 1 extra life) can change this, though.
Avoid the Ankh at all cost since it respawns you as another character and would therefor void the achievement.
So just pray to the RNG God to be lucky with drops and this is going to be a breeze. Time consuming but easy.
Good luck everyone!
If you vote negatively please state why in the comments so I can improve my solution. I know some people prefer going for this legtit but this is for gamers like me that don't have enough skill to attempt it that way.