I had problems unlocking this achievement on Steam even though I got the mutator easily on console. To get the Wheel Goat mutator, all you have to do is knock over the golden wheel deep in Bullgrurzh. However, on the Steam version, there seems to be a small glitch. You have to make sure you have unlocked the Anti-Gravity Goat mutator first.
So go to Old Goat Mountain first; climb up the tall tower using a combination of the lava pits, Double Jump mutator , and Angel Goat mutator; run through the purple ball at the top; and then go to Bullgrurzh to unlock the Wheel Goat.