Ring of Pain

Start a run with the double candle. This will let you choose a head slot item to start with.
Then play a run as normal, but look to get strong without spending too many souls and even look for ways to boost your souls, like optional levels or the siphoning shield. You will need a lot of souls for re-rolling items to get many different head slot items. Always worth getting the gold items that make mimics have an extra choice, or make the first re-roll free. Also look for the merchant mask to make chests cheaper.
Look out for the Ravaged path. It is a green door with a monster head symbol. You can find much loot here and a guaranteed head slot item
Look out for free chests, and try to search for hats in all of the 4 colours of loot to increase your odds.
I didn't get my eighth hat until quite late in the game, meaning you can't do too many re-rolls to get hats quickly. I found that unless I picked up other useful items, I would die before the re-rolls gave me my eigth hat. Basically, keep rolling if you get trash items, and take the good ones when they appear, even if they are not hats.

Also note, this seems to be related to the helmet items on the left of the inventory. I believe the mask items do not count towards thsi achievement. Any helmet slot item will do, be it a tiara or a crown