For this achievement you need to mount a cockatrice. It's straightforward enough and you can get unlock this achievement very early on in the first level.
Cockatrices are pink and yellow creatures that the enemies will occasionally appear on throughout the game. To mount/ride one you'll first need to knock the enemy off, do so by attacking them, this can be by using your melee attack or by casting a magic spell. Once you've knocked the enemy off just walk over the Cockatrice and your character will automatically mount it, unlocking the achievement.
The first Cockatrice of the game is encountered very early during the first level. Once you encounter it and are able to mount it you should do so promptly, not doing so will allow the enemies to mount it again or worse, the Cockatrice may flee. Below I've linked in a video which will show you where to expect it, it appears at the 40 second mark.
* Credit to WeakestWastrel for the video