The following achievements will unlock automatically as you progress through the DLC's main story. Finish the campaign and you'll acquire a total of 5 achievements.

Stone Cold Killer
Complete mission “Blood from a Stone”.

It's a Wash
Complete mission “The Name of the Law”.

Getting Off Track
Complete mission “Off the Rails”.

Bounty: Hunted
Complete mission “Where It All Started”.

Ruin on the Horizon
Complete mission “Riding to Ruin”.

Helpful Hunter
Complete all side missions on Gehenna.

There are a total of 11 Side Missions scattered throughout the DLC maps; some of them will only become available once you've progressed far enough into the main story. None of the choices you make during these quests will impact the achievement.

Vestige

• The Quick and the Quicker — I've heard the quest glitches out for some players, so you ought to complete this one as soon as you get it.

The Blastplains

• Miracle Elixir Fixer
  
• Money Back Guarantee — If there are nearby enemies in the area, clear them out before entering the man's shop. Otherwise, the enemies will follow you into the store, which will scare the quest giver away from the counter and you won't be able to talk to him. If this happens, simply exit the map and return right after.
  
• Devil Rustlers
  
• Saurdew Valley — Avoid killing the Imps as the quest might bug out on you. For me, the quest indicator disappeared from the map and I couldn't find any other eggs afterwards. Despite this, if you complete the other two assignments and turn them in, the side mission will count as completed for the achievement, however, the mission itself will still remain active.

Ashfall Peaks

• The Legend of McSmugger
  
• Dirty Deeds

Obsidian Forest

• Of Blood and Beans
  
• Lost and Found

Bloodsun Canyon

• The Meatman Prophecy
  
• The Dandy and Damsel

That's Quite Challenging
Complete all crew challenges on Gehenna.

Some Crew Challenges won't be accessible until you've progressed far enough into the story to unlock new abilities (the bouncing green shrooms, teleporters and the ability to shatter explosive crystals). Keep in mind that Sato's Cache drops can only be obtained after you've collected all of his Sagas on each map. Once collected, the cache drop's location will be marked on your map.

NOTE: There's a chance that some Good Prospects chests might not count the first time you open them (I believe this is caused by scripted dialogue scenes preventing the Prospector's audio logs from correctly playing). If you run into this glitch, simply exit and reenter the map. The chest should now reset, allowing you to reopen it once again, completing the Crew Challenge.

Color Map Guide:

• Red circles - Sato's Saga
  
• Pink circles - Sato's Cache
  
• Yellow circles - Creature Feature
  
• Green circles - Good Prospects
  
• Orange circles - Most Wanted
  
• Blue circles - Skin to Win
  
• White circle - Sheriff Prison Cell Chest — This is an optional bonus chest on Vestige and is not required to get any achievements. You must first find all other Good Prospects chests before this one appears. After which, go to the Saloon and find a bookcase to the left side of the bar counter. Simply interact with it and you'll open up a secret door that leads to the Cell's chest.

Vestige
The Blastplains
Ashfall Peaks
Obsidian Forest
Bloodsun Canyon
Crater's Edge

In case you need a more detailed walkthrough for very specific or hard to reach locations, be sure to check out this video guide (each Crew Challenge has a timestamp). Credit for the video goes to AchievementSquad.

How the West Was Done
Complete all plot missions, side missions, and crew challenges on Gehenna.

This will unlock automatically after you've completed all other tasks on Gehenna and acquired the other 7 achievements.

Hope this helped you out. Feel free to give feedback or correct me if I've made any mistakes.