You'll get D-Horse immediately after arriving in Afghanistan and she'll most likely be the first buddy that you'll get the bond maxed out with first.
Now this is quite an easy achievement to get, it just takes some time to do. Please note that this is a similar solution that I'm posting in the other two bond achievements as well.
Now how do you increase the bond of a buddy (sans D-Walker as it is a robot)? Well you do main missions with them as well as side-ops running around the overworld of Afghanistan and Africa. Now you can see their progress at the end of main missions as well as at the sortie screen. IF in fact you do have the meter all filled up but the achievement didn't pop because you got all your buddy progress (or got the last bit needed by doing a side-op) you need to redeploy and then extract yourself inside of freeroam via helicopter, do not just return to ACC after you hit a checkpoint because the achievement will not unlock. Once the game gets back to the loading screen the achievement should unlock.
Do note that you can decrease the bond if you get your buddies "killed", as they will be extracted, by either enemy fire or friendly fire primarily from explosives.