The N. Verted version of each level will open as soon as you complete the normal version of a level, so there aren't really any special requirements.
This mode will typically offer a mirror image of the level, along with a visual and/or gameplay tweak. The style of the tweaks will be different between worlds. For instance, all levels within the Salty Wharf dimension start off as grey-scale and colour splashes out onto the background and NPCs as you spin or slam throughout the level, while your character movement in the Sn@xx Dimension levels changes to feel like everything is happening underwater.
This can be challenging for two reasons. Firstly, if you're like me and you got through the normal versions of the level by memorising timing and layout, the mirror image factor will probably throw you off. All jumps to the right are now jumps to the left and vice versa. Secondly, the visual tweaks can occasionally make things like distances harder to judge when you're jumping between platforms.
For this achievement to unlock, you will have to complete the N. Verted version of all the main levels, as well as the Tawna/Dingodile/Cortex branches that pop out from time to time.
For anyone relying on pure reflex to get through the normal version of each level, this will be just as potentially frustrating as doing the normal level. For those relying on memorisation, this can prove to be more frustrating, because you're not only dealing with the mirror image layout, but the visual/gameplay tweaks can throw you off during those carefully timed jumps.
My only advice is to persevere and to re-memorise the layout of the levels as you go along. It didn't take me too long in comparison to finishing the normal levels, but that could be because I already knew many of the levels from my initial playthrough. Good luck and good gaming!