Minor Story Spoilers
You are only able to acquire Quiet during Mission 11.
Once she's at Mother Base, return there after each mission, until you get a cutscene where Miller stops her from accompanying you on a mission. This happened to me after completing Mission 12. Complete a few more missions (for me it was after completing Mission 14), and Ocelot will ask you to come back to base. After you talk to him, Quiet will now be an available buddy.
Once you have her as a buddy, simply take her with you on every mission, side op, or free roam you do. As with the other buddies, using her will increase her bond with you. Once you max her bond rating, the achievement will unlock.
You can check your bond level with your buddies by highlighting them on the Select Buddy option in the Mission Prep screen, or upon completing a Mission on the bottom left of the ranking screen.
To use Quiet's scouting and overwatch abilities, go to the map screen on your iDroid and press up/down on the d-pad until the "Scout" or "Attack" option is chosen. It works similar to the helicopter extraction points, as you can only have her go to predetermined points on the map. As you increase her bond, you will unlock more commands and equipment to use with her. To use her commands, hold
and use the
to choose a command. Click the
to issue that command.