First of all, for this achievement you're going to need to have a Cluckles piñata in your garden, the only way to get a Cluckles is to purchase one from the Paper Pets for 1,100 chocolate coins. However you need to be at least level 8 before you can purchase a Cluckles.
Once you have a Cluckles the achievement is easy enough, you'll first need to mate 2 other piñatas, it doesn't matter which piñatas you choose to do this, the outcome is the same. Once you've mated a pair of piñatas wait a few moments and an egg will be delivered, now select the Cluckles by pressing "A" on it and then direct it to the egg, the Cluckles will approach it and then jump up and down a few times, the egg will hatch and the achievement will unlock.
An example of this can be seen in the video below.
* Credit to AnInsaneGuy for the video