The achievement requires that you not be able to fill any boxes after you first receive your dice, but that you then fill all of the boxes anyway. It's a tough achievement, largely because it requires rolling so terribly.

The Caliana/Mage combination is excellent for this feat, because there is only one red or yellow die. So, all you need to do is roll a 1 or a 2 on the red die, and you won't hit any boxes on either the rat, the rogue, or the goblin (or the phantom, if you're super lucky). Start at Veteran, because the extra black die would make it harder to miss. Dragon's Cave is easiest.

Then, stock up on potions, abilities, and blue items. I have the Heroism potion, which is amazing, but it's not necessary. Any collection of abilities and potions can help. Rogue is actually the hardest of the three. Between those abilities, Faerie Fire, and double Force Bolt, you'll have a good chance of filling all of the boxes.

Good luck!

Below is a before-and-after image of my solution.

(Apologies for originally posting this under "Artwork". It's my first time making a guide for Steam.)