So this achievement is easier to do if you first complete the Board Hoarder achievement. See my solution below.
Now, I would suggest not even attempting to go for this until you have played for a minimum of 8-10 hours, upgraded your sack, gathered a fair amount of gold (10K+) and expanded the shop keeper items at least a few times so there is a larger selection for each run. At this point, you should be able to travel further down into the mines more quickly and get to a point where you can pick up 3-4 bombs per level on average and sometimes as many as 8-10 if you find a few "Bag O' Bombs". You need to keep all bombs so do not use them for secret rooms or killing enemies. As you are travelling down through the mine visit the shop keeper each time (I skipped it a few times as required a bomb to access the area) and look out for a potion called "Doubling Saison". This potion basically doubles your bombs. So buy the potion and keep it on you until you have reached 50 and then go ahead and use it to double your bombs to 100.
As Sherlocked4evr mentioned below, "Popcorn" relic which is fairly easy to find and has a chance of duplicating an item is good and "Pocket of Holding" relic would also be very helpful to your run as it drops two bombs in a new room. Reveal spoiler on where to pick up the "Pocket of Holding" relic.
Once you have unlocked the achievement keep going to the shop keeper on each level. You now need a potion called "Impish Key Bomb". This swaps your bombs and keys around. So if you have 99 bombs when you take the potion you will then have 99 keys and the "Key Hoarder" achievement will pop.