After finishing the mission "Zero Saints Thirty", you are able to customize your Boss as you see fit. After selecting the Customize option, the second to last option is "Character Gallery". Select it, choose "Save Character" and then the slot to save it to. Once the save is complete, achievement unlocked.
You can also go to the Character Gallery > Save Character while using any of the character editing stores in the game (Planet Zin, Image as Designed, Rusty's Needles, etc.), or through the Ship Functions menu options in any of the Gateway portals. (Many thanks to BraveZach for the reminder!)