1) Do "Clearing the Forest" and find the Bloody Cloth.
2) Equip it, and get the quest from the sheriff.
3) Go to the quest giver "The Property".
4) There you get a picture of the murder victim and 5 new quest givers, now equip the photo and talk to all of them.
5) This is IMPORTANT: When you go to "The Dancer", you get to choose 1 of 2 things. These are unknown - the left one is for FEMALE only, it's the parasol for the Dancer Set. The right one is for MALE characters only - this is the hard to get, "Walking Stick" for the gentleman set.
6) Now, after talking to all of the suspects, talk to the sheriff again - 500xp.
7) It's time to gather evidence now. These are found through 6 different jobs: NOTE - EQUIP THE PHOTO!
- Breaking in Horses
- Transport ammunition
- Charlatan
- Picking tobacco
- Sawing wood
- Picking Agaves
8) After finding all these pieces of evidence, talk to the sheriff again. Now you need to pick out the murderer from all the 5 people you talked to. ( Mr. Cartwright ARE MURDERER )
It's no fun if I tell you who it is though - but 2 of the things you found should give you a clear picture of who the murderer is

You get 1500xp from choosing the right person as the murderer. Choose wrong, and you get nothing.
9) If you picked right, and got your 1500xp, go duel that person, for an additional 1500xp and a rusty colt buntline, worth $2687.
So in grand total:
100xp from talking to people
500xp from telling the sheriff what you learned
3000xp if you choose the right killer and win the duel
= 3600 experience points
Rusty Colt Buntline, worth $2687 if sold
And of course, the Walking stick or Parasol for either of those sets, depending on your character's gender