A little ways into the second chapter of the game you obtain the Substance Mobile Printer, or SMP. As part of the tutorial on how to use it you will be tasked with "printing" a dog and a cat. Open up all the display cases to retrieve the blueprints for the dog and cat as well as the material you'll need to actually print them. Once that's done, simply print 4 cats and two dogs (or 5 cats and 1 dog) onto the brightly colored platforms.
The SMP has the ability to reclaim objects that you print, however, the animals cannot be reclaimed. To get the achievement
, you can either start a new playthrough (which won't take very much time) or you can exit to the main menu, go to Chapter Select, and start Chapter 2 again.
BE WARNED: Starting a chapter over will erase your previous save data.