This stranger mission tasks Arthur with taking taken the Wanted Posters of various criminals so that they can leave town without being noticed. All the five posters that have to be taken down are all found close to the mission starting point, so this is definitely quicker than doing other missions that see Arthur having to go all over the map to complete them.
The "The Ties that Bind Us" Stranger mission becomes available in Chapter 4 after completing "The Joys of Civilization" Story Mission, which is the very first mission of the chapter. Once that mission has been completed, the "The Ties that Bind Us" mission can be started by searching for the two wanted men to the south of Rhodes, right below the town's southernmost train track.
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Once approached, the two will ask Arthur for his help. They want the outlaw to take down five wanted posters that can be found in Rhodes so that they can make their escape unnoticed.
Chain Gang Wanted Poster #1
The first Chain Gang Wanted Poster can be found very close to the mission's starting point. Head into the Rhodes station and check the board close to to the station's entrance. Interact with the object to check out the poster and remove it.
Chain Gang Wanted Poster #2
The second Chain Gang Wanted Poster can be found not too far from the station. Head into town and you should notice a man leaning on a pole. The poster is on the same pole. Politely trying to get the man to move for a second so that Arthur can take the poster down will amount to nothing, so antagonize and take him down to finally be able to reach the poster. Avoid looting the downed man, as you will likely be reported to the law.
Chain Gang Wanted Poster #3
The third Chain Gang Wanted Poster is found near the Sheriff's Office, it's on the board nearby. Just check out the object and take the poster down.
Chain Gang Wanted Poster #4
The fourth Chain Gang Wanted Poster is found on a lantern post at the end of the Rhodes main road. Just wander the edges of town for a bit and it will be highlighted on the map.
Chain Gang Wanted Poster #5
The fifth and final Chain Gang Wanted Poster can easily be found right past the fourth one thanks to a bounty hunter who has actually already taken it down. Arthur can ask the man to give the poster to him politely, but that won't work....
Instead, you can beat the bounty hunter or use the Lasso to capture him and then loot him, but always make sure there are no witnesses around. The lawmen in Rhodes are aware of the Van der Linde gang's activities in Chapter 4, so they will come out in droves to take Arthur down.
Rewards and Endings
Once all posters have been taken down, you need to get back to the location of Mr. Black and Mr. White to complete the mission. Here, Arthur will ask more about what they have done, as the bounty is pretty high for some common criminals.
Once he has heard their explanation, you can make a choice: burn down the posters and let the two wanted men go, or bring them to the law. The latter choice is the most profitable, as you can cash the bounty they have on their heads.
Ending 1 - Turning Them In
Once (and if) you have decided to play as Bounty Hunter, you need to act fast. Activate Dead Eye to capture the first wanted man, then whistle your horse to your position to capture the second one. As you can only stow one man on the horse, just carry one yourself and make the horse follow you to the Sheriff's Office. Once there, put the two in the cell and interact with the Sheriff to receive the money.
Ending 2 - Help Them
Should you decide to burn the poster and help Mr. Black and Mr. White, you will continue to meet them for a couple more times.
After burning the poster you will encounter them just north of Rhodes town. They will be in a pretty bad state and in need of medicine. You'll come across them just near the letter "L" of Lemoyne on the map and they will ask for medicine. If you have some on you, then give it to them and the encounter will be completed successfully. If you don't have any medicine on you, you can always ride to the nearest town and buy it from the general store.
After you give them the medicine, Arthur advises them to go far north and separate if they wish to evade the law. Taking his advice, they both move north to the Big Valley area, near Hilt Shack. Later on, they can both be found living in a tree house and in a better shape. You can find them at the end of Little Creek River near Hilt Shack. This will be your last encounter with the pair.
Once this is done, the "The Ties that Bind Us" is complete.