This achievement can be unlocked once the player gets to Blackwater. There will be a herd of 20 buffalo wandering the Great Plains area (there is a picture of buffalo on the map in this area). The player must kill all 20 buffalo in the herd to get this achievement.
Since the buffalo run when attacked or if the player approaches too closely, attempt to kill them from a distance, using head shots from a sniper rifle. Done right, few of the buffalo, if any, will run. If they do run, do NOT chase them. Simply leave and return at a later time (1-2 days in game) and repeat the process until all of them are dead. Giving chase could force one or more of them off of a cliff, or into the water, or even onto the train tracks in front of a train. If this happens, the dead buffalo will not re-spawn, and the player will not be able to kill it, making the achievement impossible to unlock in single player until a new game is started.
In this unfortunate situation, there is a way to get the achievement in multiplayer. If you can find someone who has reached Legendary Rank 3, Level 35 in multiplayer (no mean feat, since getting each Legendary Rank requires a player to reset at Level 50 and level up all over again), they will have unlocked the buffalo mount. If this person can meet you in game, they can call for the mount and you can kill it. Do this 20 times and the achievement is yours. The whole process should take no more than about 15 minutes. As far as I know, there are no buffalo otherwise available in multiplayer.