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 to use phosphorous coating on your bullets and go into dead eye and kill 5 Undead (although this rarely works). This method works and you will progress to the next rank. To make sure it works, make sure to use a repeater and empty the clip.
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.