This achievement is difficult, especially since the buffalo scatter and run when attacked, thus disappearing if they get out of Marston's sight. The best method to use is to throw dynamite into the herd, and finish the rest off using Dead-Eye.
However, if the herd scatters out of Marston's sight, you can come back at a later date and continue to kill the buffalo with each kill still counting towards the achievement - you don't have to get them all at once!
It's worth mentioning that there are a total of 20 buffalo; if you're not sure how many are left, go to Pause > Stats > Misc and see how many buffalo are left.
Tips and Tricks
One good tactic is to approach the herd by horse, take the Evans Repeater and some Moonshine and use Dead-Eye to paint all the outside buffalo. After this, go straight back into Dead-Eye (where weapons get automatically reloaded), and shoot the rest. Do this one more time and then chase the rest of the buffalo.
Another approach would be to stock up with Moonshine and other Dead-Eye restoring items, and a full set of throwing knives. Try to "herd" them in one direction (as they will more than likely run from you), pick off likely stragglers first, then work your aim inward. Be close, aim right. Using Dead-Eye with the knives will kill them all in one shot each time. Once finished, you may have lost a good number of the bodies, so if you want to skin them all, reload and then proceed as normal.
Another idea: shoot the buffalo with the Bolt Action Rifle while on horse back. It would be smart to skin the buffalo because everything is very high valued. These skins and meat will pay for the bullets and Dead-Eye needed to kill all of the buffalo.
If you wish buffalo to remain in the game but still want the achievement/trophy, and you don't mind not earning money from buffalo resources, then simply save the game before killing the buffalo. Go and kill them and gain the achievement, and then reload the game; the trophy/achievement will still be present as will the buffalo. Players can also use this method to skin the buffalo after gaining the trophy/achievement so that they don't have to worry about skinning them all while trying to stay on the herd at the same time.
This is a good achievement to take on over time. Use a Carcano Rifle and Dead-Eye from a long distance. The rest may scatter but will regroup with time. Then as you're passing by the next time, repeat the process. It's possible to Dead-Eye almost all of them at once with an Evans Repeater, but it has to be headshots.
It is also possible to get this trophy/achievement while playing online Free Roam by killing 20 buffalo. All that is needed is either to have unlocked the buffalo as a mount or have someone else who has already unlocked the buffalo.
Don't rush to kill all 20 buffalo, just kill some you can reach quickly. Let the living go and skin the dead. A good method to maximize profit is to kill and skin as many buffalo as you can while wearing the Expert Hunter Outfit, then change to the savvy merchant outfit, finally sell the meat, hide and horns at Escalera (the provisions are worth more when you have killed all 20 buffalo).
"Manifest Destiny" is a reference to the belief during the 19th century that the United States was destined, and had the God-given right, to expand across the North American continent and take everything as their own. The reference is to the US military that tried to hunt the Buffalo to extinction to starve out Native Americans and allow for westward expansion. The idea of a coast to coast country was called Manifest Destiny and was used to support the war with Mexico. In this war, the US annexed from Mexico parts of Texas along the disputed Republic of Texas border (south and west Texas and the panhandle), part of present day Colorado and Wyoming, and all of New Mexico, Arizona, Nevada, Utah, and California.
One of the Vice cases in LA Noire another Rockstar game is also called Manifest Destiny.
Nastas also claims that the Buffalo will soon be wiped out by hunters, to which Harold MacDougal says is preposterous, claiming that the Buffalo will adapt to the hunting and grow stronger over time.
If you've already killed 19 Buffalo and kill the Buffalo that appears in the mission "At Home With Dutch" you can get the achievement and still go back to the Great Plains and kill the 20th Buffalo.
Not actually a glitch - some players have experienced failure of this achievement due to the buffalo supposedly "Committing suicide". This means that, instead of the kills counting towards the achievement, one or more of the buffalo have run into a river or train track/fallen off a cliff. If this happens, the achievement will be impossible to get, and the only way to claim your 5G/Bronze award will be to start the game again or kill the buffalo online. So, it is recommended to save before hunting to avoid this.
It is possible to kill 21 buffalo after completing the achievement/trophy. One may encounter another hunter chasing a buffalo; kill this buffalo before the hunter does and skin it. The hunter gets rather irritated but it counts as another kill in the buffalo tally.
Laying a "trap" in Tall Trees can summon a small herd of buffalo (5) after already receiving the Achievement, so player can get 20 buffalo kills and above in this glitch.