Note that this guide was written from the perspective of someone that already completed the main story line and hideouts in the game. You can work on these challenges while going through the story, if you haven't finished yet, and you probably should, as it's a little less boring.
Throughout these challenges you should also remember to loot the body of every enemy to retrieve your tomahawks. It's also a good idea to save after each completed rank. I chose the safehouse in Armadillo since the Twin Rocks Hideout was my main base of operations.
Similar to the ambient challenges in the base game, there are a few tomahawk challenges included in this new DLC. Unfortunately there are only five ranks for you to complete this time. You can see what your next challenge is by pausing the game and selecting your journal.
Rank 1: Purchase a tomahawk at the general store in Manzanita Post.
It doesn't get any easier than this. Call your horse, set a waypoint for the general store in Manzanita Post (located in the top right corner of the map south of the Tall Trees area), and ride. Each tomahawk costs $30 and your limit is twenty. I would recommend buying the limit, if you can afford it.
Rank 2: Kill one enemy with a thrown tomahawk, one with a tomahawk melee, and one while on horseback.
While on horseback, throwing the tomahawk is pretty much your only option. For some reason that does not stack with the throwing tomahawk kill though, you have to do one on foot and one on horseback.
I went to the Twin Rocks Hideout for this one. It's very simple and only has a few enemies. Ride up to the low wall out front, and whip a tomahawk at the closest enemy. Jump off your horse, target another enemy, and hit him with a tomahawk. To melee an enemy, get up close to them, hold left trigger and then hit right trigger. It will automatically melee instead of throwing. You'll also continue to hold on to the tomahawk.
Rank 3: Kill a flying bird with a thrown tomahawk and no dead eye.
I thought this was going to be a lot tougher than it was. Birds typically fly too high for you to throw your tomahawk at. I started looking around for birds that were sitting on a post, and was going to wait for them to fly away and nail them, but that was taking too long. Luckily i found a spot that works well for this. There is a wooden bridge that extends between two dirt hills, over a path. It's right above the first "n" in Macfarlane's Ranch in the normal view of the map. If you're by your safehouse in Macfarlane's Ranch, and set a waypoint to tall trees, it will take you right under it. Climb up one of the hills and get on the bridge. The birds fly right by you, at about eye level. Just wait for one to get close, lock on, and throw.
Rank 4: Kill five enemies in a row with the tomahawk without missing.
Head to the Twin Rocks Hideout again. Take out the gang members on the tall rocks first with a normal gun. Now there should be four enemies left outside (one on the roof and three on the ground). Take them one at a time. Get at a good angle to an enemy (no obstructions/cover in the way) go into dead eye and mark him. The tomahawk is pretty much an instant kill, so don't worry. Once you eliminate the four members outside, at least one will run out of the front door of the house. Do the same to him.
Rank 5: Clear a gang hideout using only tomahawks.
This is not too hard. I had to do it three times though due to uncontrollable circumstances, which I will post for you to avoid.
My first attempt was at Twin Peaks because there's only a dozen enemies. There seemed to be some sort of glitch that would not allow me to kill the gang member on the rock to the left with a tomahawk, no matter how many times I hit him.
My second attempt was at Fort Mercer. I completed the hideout using only my trusty tomahawk, but the rank did not complete. I believe this is because i let Leone in and he uses a gun. I didn't retry without letting him in.
Instead I went to Tumbleweed. There are a good number of gang members to take out, but the buildings in the town and the low walls around the mansion offer good cover. Take it slow and don't run out in the open if there's a lot of enemies. On this rank it's more important than ever to loot the bodies and get your tomahawks back so you don't run out.