You get a boss stem cell the first time you complete the game, and another for each time you complete the game with another active stem cell. The first stem cell, when turned on, activates hard mode. Stem cells can be turned on from the starting room by standing next to the long glass container.
I find that the difficulty of the enemies and bosses is about the same on hard, if not exactly the same. More advanced enemies do spawn earlier, and you get fewer health refills, but if you got to hard difficulty, and you have 4 potions, you can handle the early levels no problem.
The main roadblock to this achievement is that the health refill does NOT appear before the final boss, which means you need to do the whole castle and final boss on the same potion set.
As you probably know, those castle enemies are very deadly, and hit you through the walls and floors if you're not careful. The time I got this achievement, I had a freezing bow, and I used it constantly in between using my other skills so that enemies couldn't react. Any time you drop down a level, roll away immediately in case a spear enemy is below. When you do the key rooms, consider turning around after killing the elites, as weapons or upgrade items seem to never appear in those rooms; no sense risking killing more enemies just for more cells.