Red Dead Redemption

I found this easiest to do with "Political Realities in Armadillo", the first mission from Marshal Johnson. You can replay this at anytime from the pause menu. Just go to stats, then mission and select it. There are three categories that you must do well in to get a gold. (Time, Accuracy, and Head Shots). These three DO NOT have to be completed on the same run through of a mission so you can replay it several times.
Here's how to accomplish each category

Time: Must be 2:34 or lower.
Skip every cutscene with the "A" button
When the mission starts tap "A" repeatedly to run ahead of the marshal to the saloon.
After skipping the scene, whistle to your horse and get on it as fast as possible and then race to Pleasance House, the marshal will keep up.
Shot everyone as fast as possible (use Casual targeting mode) and I found it fastest to just shoot the main guy in the head instead of the leg.

Accuracy and Headshots: 100% and 5 headshots
These can be done on the same run through. Very simple, just make sure that you use Dead Eye to line up the headshots.

*Edit: Thanks to everyone who confirmed the time.
**Edit: Changed time again (lol) to 2:34 or lower. Thanks again guys for the times and all the positive feedback.

I found this easiest to do with "Political Realities in Armadillo", the first mission from Marshal Johnson. You can replay this at anytime from the pause menu. Just go to stats, then mission and select it. There are three categories that you must do well in to get a gold. (Time, Accuracy, and Head Shots). These three DO NOT have to be completed on the same run through of a mission so you can replay it several times.
Here's how to accomplish each category

Time: Must be 2:34 or lower.
Skip every cutscene with the "A" button
When the mission starts tap "A" repeatedly to run ahead of the marshal to the saloon.
After skipping the scene, whistle to your horse and get on it as fast as possible and then race to Pleasance House, the marshal will keep up.
Shot everyone as fast as possible (use Casual targeting mode) and I found it fastest to just shoot the main guy in the head instead of the leg.

Accuracy and Headshots: 100% and 5 headshots
These can be done on the same run through. Very simple, just make sure that you use Dead Eye to line up the headshots.

*Edit: Thanks to everyone who confirmed the time.
**Edit: Changed time again (lol) to 2:34 or lower. Thanks again guys for the times and all the positive feedback.

I had a hard time getting my gold medal in "Political Realities in Armadillo", so I tried "Old Swindler Blues" and I got the achievement on my second try. Since you can retry few times to get the gold medal, one walkthrough you go as fast as you can without killing anyone (you should be able to do it in under 2:00), the second time you use your dead eye and head shot the 4 enemies that attacks you and just enjoy the ride to Armadillo. Ding ... achievement won. :)

I made this video based on (Doctor Caligari guide).<-credits ...

Political Realities in Armadillo.
the first mission from Marshal Johnson.
Mission time: 2:30
Shot Accuracy: 100%
Headshots: 5
hope it helps

I have just found out that this achievement is cumulative, you do not need to get the fastest time, shot accuracy and number of headshots all in the same playthrough.

I tried the mission "Political Realities in Armadillo" just then and got 100% accuracy and 6 headshots but finished it in 2:39 seconds, then when i played it again i did not headshot anyone and i missed alot of shots however i got the achievement. When i checked in missions it said i finished the mission in 2:18 with my 100% and 6 headshots still in tact

I hope this helps.

first, go to the pause menu, stats, missions and check which mission you have have Silver on. if any missions are close to the time/accuracy/headshot requirements, i'd say go for that mission to gain this achv.

what mission worked for me:
Old Swindler's Blues
Mission time: 2:11
Shot Accuracy: 100%
Headshots: 4

you DO NOT have to meet all 3 requirements time/accuracy/headshots in one run.

first runthru, i ran through the mission as quickly as possible, only shooting as needed (damn, its hard to steer and shoot, all while listening to Dickens whining the entire time >.<).

then the second time i ran through the mission i shot and killed everyone for 100% accuracy, using dead eye as needed, but had problems steering so i never made it to armadillo in time and the old bastard died on me.

my third runthru, i went for headshots. you get 2 waves of 2 enemies. i simply stopped the wagon both times, dead eyed and boom, headshot.

seems like a cheap way to get a gold medal when i'm sure there are plenty of people talented enough to attain the medal legit, but rockstar is good about making the achvs fun and attainable for everyone!

hit me up if you need help: geegers78

first, go to the pause menu, stats, missions and check which mission you have have Silver on. if any missions are close to the time/accuracy/headshot requirements, i'd say go for that mission to gain this achv.

what mission worked for me:
Old Swindler's Blues
Mission time: 2:11
Shot Accuracy: 100%
Headshots: 4

you DO NOT have to meet all 3 requirements time/accuracy/headshots in one run.

first runthru, i ran through the mission as quickly as possible, only shooting as needed (damn, its hard to steer and shoot, all while listening to Dickens whining the entire time >.<).

then the second time i ran through the mission i shot and killed everyone for 100% accuracy, using dead eye as needed, but had problems steering so i never made it to armadillo in time and the old bastard died on me.

my third runthru, i went for headshots. you get 2 waves of 2 enemies. i simply stopped the wagon both times, dead eyed and boom, headshot.

seems like a cheap way to get a gold medal when i'm sure there are plenty of people talented enough to attain the medal legit, but rockstar is good about making the achvs fun and attainable for everyone!

hit me up if you need help: geegers78

The easiest mission to do this achievement in is "Political Realities in Armadillo". The gold medal in this mission is determined by theses three things:1.Perfect accuracy
2. 5 headshots 3. Beat the mission in around 2 mins 35 secs
*They do not all have to be done at the same time*
A good method would be to do the time one first.To do this, simply skip all cut-scenes, and run as fast as you can. When you have to kill enemies, do it efficiently with dead-eye. After that, take your time doing the perfect accuracy and headshots by carefully using dead-eye.
*Remember that you can replay missions through the mission select in the main menu*
Cheers!

I did this on "A Frenchman, a Welshman, and an Irishman".

Just dead eye headshot the Frenchman and Welshman and ride out to the cabin on the lake. It's useful to pause and mark it on your map. It's hard to get much separation between you and Irish before he goes into the scrub to meet at the cliff because every time you're in front of him and hold down A it slows you down to match him, but you can still get there quickly.

Ride down and be ready to go into dead eye as soon as the shooting starts. (You can stay on your horse at first for the high ground over cover, or you can dismount. I prefer dismounting because it puts all of them at the same level so I don't accidentally paint somebody's chest or arm when trying to get someone else's head.)

Go into deadeye and paint heads, but do it quickly. For a good time I recommend you use the repeater and paint as many targets as possible before they run to cover. If you're quick you can pretty much get them all, though you may not get 100% headshots. If not that's okay, just consider your first run the time run and keep going. Try not to use up all your dead eye so that you can use it to quickly pick off any survivors as well as to headshot the guy that comes out of the door at the end.

When marking headshots I've found it works great to simply stand on foot aim to the left or right of someone's head then sweep horizontally while in dead eye, just be sure to do it high enough not to get a shoulder or anything. This works great for marking multiple targets since their heads are often at the same height when you're on foot, but be careful because they will not be exact due to them taking cover and various things. If you have to come in from above instead and just bring the aim down on top until you get a mark. Just be sure that you stop as soon as you mark the head or you'll mark something else.

Now if, like me, you managed to get a great time, but accidentally painted a few non headshots in your rush don't worry, it's even easier to get your headshots if you aren't in a rush. For a pure accuracy run dismount your horse early to keep it out of the way and walk up until they start shooting, or are about to, then dead eye and quickly paint heads, but only sure headshots. If you mark a shot that isn't a headshot cancel your deadeye. Get as many as you can before they break for cover, but don't worry about getting them all. Do NOT use up all your deadeye. In fact it's perfectly okay to merely mark 2 or 3 shots right at the start.

Now for the remaining enemies you will wait for them to pop up out of cover and dead eye headshot them one at a time. It uses very little dead eye to do this so you shouldn't run out. Since time is not an issue go ahead and take cover as long as you can get a clean shot on their head when they pop up to shoot. If you cannot get a totally clear shot at their head do not even try to use dead eye because a wasted shot will ruin it. Just move around until you get your chance, take your time, and don't forget to move the aim up after it snaps to the target BEFORE you enter dead eye. (You can easily get a few headshots without dead eye, but you also might miss if they choose to move right when you pull the trigger. With dead eye even if they do move as long as they don't move their head behind cover the shot will still land perfectly, so dead eye is more of a sure thing.)

Don't forget to headshot the guy as he comes out the door. And remember it's okay to cancel a dead eye, even if you take a hit. Damage taken does not factor into your ranking only accuracy and time.

There are more enemies than "Political Realities in Armadillo", but it's still just a 2 minute or so mission and at least you have access to dead eye level 2.

This can be done easily along with the Gunslinger Achievement:

Start the "Political Realities in Armadillo" mission.
Run through as fast as possible by not waiting for the Marshal. Skip the cut scene. Then follow the Marshal on horse to the hideout. Shoot all 5 enemies in the head using dead eye (This will trigger the Gunslinger Achievement if you are using Expert targeting). Then gun down the leader.

If you get 100% accuracy and 5 head shots in under 2:30 the gold is yours.

None of the other solutions here mention what I consider to be the easiest and quickest way of doing this achievement.

Replay the mission "Lucky in Love", skip all cutscenes, go all in twice in poker to make that go quicker, kill the guy in the duel, kill the other guy when he takes the woman hostage. Take cover behind the wall and kill the 4 remaining people in one use of dead eye and that's all you need. Make sure to get head shots in each use of dead eye, it's all over inside 3 minutes.