You'll need the Positronic Brain breakthrough. Since there are 60+ possible breakthrough technologies, you can only discover 20 of them in any one playthrough, and you have no control over which ones you discover, it may well take you several playthroughs to discover it. Note that the Creative Biorobitics (biotech) research and the Biorobitcs Lab building have nothing to do with this achievement; the positronic brain is the key.
You must also research Drone Printing on the robotics column and build a Drone Assembler building. Once you have this building completed and staffed and you have researched the positronic brain, simply order a biorobitic colonist in the assembler, and when it's finished, this achievement will unlock.
Don't forget about: