This can be done in Freeform, Challenge, or Campaign.
Once you become level 3, you acquire the ability to build Breeding Centers (under Facilities). Place a Breeding Center and hire a staff member to unlock this achievement.
Or you can wait and hire a different staff member later.
Note that in Campaign mode you may start above level 3, so you can just hire a Breeding Specialist, Janitor, or Zoo Keeper immediately.