You are allowed to gather coal and any resources at the beginning of the game like normal. as well as get coal from outposts.
WARNING: BUT you CAN NOT BUILD ANY COAL PRODUCING BUILDINGS! ONLY charcoal kilns. The moment you build a single non-charcoal kiln building you will lock this achievement for the remainder of your playthrough
Recommend: Easy difficulty, and you can get a few other achievements while you aim for this one. (Better Than London/Worse Than London)
Research sawmill ASAP then research charcoal kilns (to do this you need to research level 1 then that. I recommend doing this before day 4/5 as you will start to run low on coal).
Then just don't touch any other building.