Not an optimal guide in any way, but it's probably better than no guide at all.
As the achievement states, you must complete the tower without upgrading your equipment through any of the tower merchants. Straightforward, but more challenging than a normal run. The way I got it was through focusing on enchantment points in the early game, always grabbing that as a floor reward unless I felt that I was really lacking in health or damage output. I got into a rhythm of only taking a weapon, artefact or armour every four rounds or so, which left me with enough points in the end for almost fully enchanting the three legendary gear that I picked up along the way. Looking at enchantments and artefacts, I was always on the lookout for any healing capabilities or stun/crowd control, so that a weapon with radiance would be enough for me to take it even if my previous weapon's damage was fine. Floor rewards seem to be static as well, so if you find a good build and die late in the game, you should be able to get the same items on your next run.
My final build consisted of a quick radiance weapon with the mushroom and healing totem artefacts for massive passive healing, an armour with the snowball and fire aura enchantments to slow down and damage incoming mobs, and a crossbow with health potion cooldown on kill to quickly refill my potion while keeping mobs at a distance for emergencies.