Quick note: This is a cumulative achievement and does not need to be done in one game. You need to open a total of 100 doors across all your games.
In adventure mode you need to find the hidden keys lying around the levels and then take them to open any of the wooden doors. Keys can only be used in the level you find them in.
You can check your progress towards this achievement through the statistics on the right hand side of the main page.
There is normally about somewhere between 0-3 doors on each level. I had to complete 75 levels before I unlocked the achievement, which took me about 7/8 hours.
This is a fairly straightforward achievement to get, just a little bit time consuming. Let me know if you have any questions or suggestions for improvements for this guide. Good luck!
MathGuy 42 has posted another method in the comments which may be slightly faster:
'I hit Menu, then New Game until Level 1 has a monster or door near the beginning. I blast through to get the monster or door. If there are more nearby, I'll keep going. Depending on how far I am, I'll either intentionally die or finish Level 1, both of which save progress. Note that Menu / New Game does not save progress. I'm not done yet, but I think it will take a lot less than 7-8 hours.
Of course it also makes sense to start this mini-grind after getting the Level 20 achievement since plenty of doors and monsters will need to be dealt with for that.'