You can find DD as soon as you get off the chopper at the start of Mission 4. He'll be a puppy running around where you start. You should get a call when you are close to him. Fulton him away, and he'll return to Mother Base. From there, Ocelot will train him. After completing missions, return to Mother Base until he is fully grown. After that, he will be able to be used as a buddy.
Once you have him as a buddy, simply take him with you on every mission, side op, or free roam you do. As with the other buddies, using him will increase his bond with you. Once you max his 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.
As you increase his bond, you will unlock more commands and equipment to use with him. To use his commands, hold
and use the
to choose a command. Click the
to issue that command.