Weird West: Definitive Edition

During the "Protector Journey" (Across Rivers), after acquiring the Copper Gorget to be able to speak to ghosts, go to Quickbend; by the graveyard there's the ghost of a woman that's going to give you the quest "Last Request"

Go to Cordwood, take the Key to the Cellar on the nightstand, go down into the cellar and pick up the "Bloodstained journal" there.

Now go to the Festering Pond, loot the bandits for the Sigil Stone and go down the stone gate. In the room on the right, in a chest, pickup the Bloody Hammer.

Go back to Quickbend, give the Bloody Hammer to the Sheriff and voilà! You'll unlock the achievement, +10 Rep and a 4-star shotgun (Gold black mountain widowmaker)

Missable. The first time I encountered a ghost with a quest was right after I completed "The Strangeness" quest given by a couple of kids while traveling (Chapter 5). This is the quest needed for the Mystery Machine achievement. This was in the same town. After I completed it I traveled away from the town, came back, then loitered day and night until I found this ghost near the graveyard.

My earliest encounter with a Ghost with a quest was in Chapter 5 as the Oneirist. Travel to the town of "Murphy" and within the Graveyard you'll find a ghost needing help solving a murder. His quest will firstly take you to "Emilio's Stead" to collect a document, then onto the "Festering Pond" to collect the murder weapon. Once you have both items return to the Sheriff in Murphy to complete the quest.

This achievement is for completing a randomly-spawning side quest called "Last Request" that is given by a ghost hanging around a graveyard in any town. You won't be able to complete this before Chapter 3. After progressing far enough in the Protector's storyline, you will eventually get a medallion called the "copper gorget" that allows you to talk to ghosts, and from this point forward there will be a chance that the ghost quest-giver will spawn. Again, it's a random chance, so it isn't guaranteed to happen in any specific town that other people have named...you just have to travel around checking graveyards. Also be warned that the ghost did NOT have a blue quest marker over its head when I found it (but it is marked with the little moon symbol on the HUD minimap). I just saw a ghost standing next to a town graveyard and talked to it, and sure enough it had the quest.

Don't worry about transferring the medallion to the next character's inventory - your Chapters 4 & 5 characters will automatically have the "spirit walker" ability and be able to talk to ghosts. For me, this spawned in the first town I went to in Chapter 5, and I see a lot of other reports of people getting it in Ch. 5, so possibly the spawn rate for the quest gets higher when you get that far into the game. Regardless, I would not recommend wasting time going from town to town to town trying to find it as soon as you get the medallion in Ch. 3 (like I initially did ).

As for the quest itself:
The ghost will give you the name of a small farmhouse to visit. Go there, enter the cellar via the storm doors around the side of the house (will need to lockpick or find a key, possibly looted from any NPCs that might be on the property), and pick up the journal in the cellar. The journal will give you the name of a second location that you need to visit to find the murder weapon. For me, it was in an underground temple and once I got near the box it was stashed in, a little quest marker popped up. Once you have the weapon, go back to the town where you started the quest and turn in what you found to the sheriff. At this point you'll receive a reward and the achievement will pop, but you'll have the option to take a bounty on the murderer if you want lore closure.