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This achievement can be obtained in one of two ways, though you should follow the second method if you're trying to do a no-kills playthrough.

The first method, which is to consume the Altered Biocell after retrieving it from the Laundry Room of Cell Block A (the code to enter this room is 6014), is by far the quickest, but it will void the "Winners Don't Use Drugs (or Biocells)" achievement unless you make a save beforehand. Take note that consuming the Altered Biocell will grant you six Praxis points.

If, however, you want to complete your playthrough without consuming the item at all (and by extension avoiding hostile confrontation with the prisoners after the prison riot later on in the DLC), speak to Flossy in Cell Block A after picking up the Biocell. Through dialogue, you can mention you have found the item and if you agree to help him out, he will ask that you bring the item to a man named Red Shoes in Cell Block B. Before leaving, Flossy will hand over a Prison Access Card that will allow you passage to Cell Block B.

From here, make sure this new Optional Objective is selected from your mission log and follow the waypoint to Red Shoes' location in the yard of Cell Block B. The first waypoint will see you passing through Cell Block A's main gate and out into the courtyard between the two cell blocks. I highly recommend taking a pit stop at the Maintenance Shed to the east in this small courtyard. You will need a Multitool Device in order to open the locked keypad of the Maintenance Shed, though. Inside this Shed is a locker that contains a yellow Cell Block B jumpsuit. Make sure to grab it for both the "Dressed for the Occasion" achievement and so that you can walk freely through Cell Block B's yard without raising any alarm/suspicion.

Once you have jumpsuit, leave the Maintenance Shed. Your next objective is to make your way into the service tunnels directly north of your current position. Sneak around the Security Office here, avoiding the prison guards and security cameras, and dip down the stairs and through the door to the service tunnels. Now make your way down the tunnel, but make sure to avoid tripping any of the lasers; the first can passed through if you time it just right and the second can be avoided by jumping on the fuse box to the right-hand side of the tunnel.

At the end of the tunnel, jump up on the fuse box to the right, remove the grate from the ventilation shaft, and follow the shaft all the way to its end. You'll eventually reach a ladder that deposits you into the prison yard. Upon exiting, turn around, pass through the gate in the fence, and you can finally speak to Red Shoes in his isolation chamber. In dialogue, you can choose to give him the Altered Biocell, thus finally unlocking the achievement.