Undead Sharpshooter is a challenge in Undead Nightmare.
Ranks
Rank 1 - Kill 5 undead in 10 seconds using a gun.
Very simple and you will do this automatically when you progress through the story. If you find this hard then change your targeting setting to casual.
Rank 2 - Headshot 10 undead in a row.
Use the casual targeting setting and you will aim automatically for the head. Kill 10 regular zombies and you will progress to the next rank. Headshooting special zombies like the Retcher will reset the headshot counter.
Rank 3 - Kill 5 flaming undead in one dead eye meter.
There are two ways in doing this. One is to save Thieves Landing and use the Fire Bottle and light a group of Undead on fire and finish them off in dead eye, its highly suggested you try this in Tumbleweed using Undead Bait to first crowd a group together, then use the the Fire Bottle. This is the harder method.
The second method is much easier to perform, and requires the use Phosphorus Coating on your bullets. The best place to find a large group of zombies is the mouth of Jorge's Gap. Fast travel to Armadillo and then ride to Jorge's Gap. Go to one end of the cliffside, and use undead bait to corral the zombies together (make sure you are using a repeater in case you need to target more zombies in a single clip). Next, add phospherous coating to your bullets, and enter dead eye. Once you've entered dead eye, use the right bumper (R1 on PS3) to "mark" each zombie's head that you would like to kill. Do this fast before the dead eye time runs out, and then press to shoot. John will then proceed to shoot all five zombies in the head one after another.
Tip: If you are lucky enough to have found and broken the horse War, an extremely easy way to do this challenge is to drive war up and down the middle of Armadillo running zombies over as you do. No shooting is required. You can also ride War through Jorge's gap, then headshot five of the undead you set fire to, if you have the Legends and Killers Pack, and have acquired the Explosive Rifle, that will do fine.
Rank 4 - Kill one of each undead types in one dead eye meter.
For this go to El Presidio and outside the gate a group of undead will spawn. There will be the 4 types of zombies required for this rank. Go into dead eye and paint their heads and let it rip. You will be promoted to the next rank.
Alternatively, use Undead Bait and the Explosive Rifle. Stand by the canon, throw bait(s) over the wall untill there is a good gathering. Go into Dead Eye, point and shoot.
Tip: You can also use one of the several cannons in El Presidio to kill all 4 types in 1 shot.
Rank 5 - Get 5 undead sniper kills in 8 seconds (without using dead eye).
A sniper rifle is required. Save Plainview or Pacific Union Railroad Camp to get the Rolling Block Rifle. The easiest spot to snipe at a large cluster of undead is at the northern mouth of Jorge's Gap (near Odd Fellow's Rest, west of Armadillo). There's a high concentration of undead in the gorge, particularly at the opening. Headshot five within 8 seconds. Rewards: the Carcano Rifle and the Mauser Pistol.
An alternate method of completing this task (and arguably easier) is to go to Jorge's Gap, and instead of aiming to kill each zombie, run into the group of them and try to get off five "execution" shots.
Another method would also be to coat the bullets with Phosphorus so that you need not worry about headshots.