Nat J, i need your advice on HOW to cull without detailed equipment?!
Different people do it in different ways. Most killing methods have one thing in common: they use a tool the person might have handy, that also has other uses. The "best" method depends on what tool and what skills you have.
Personally, I use a hatchet or ax or similar tool to chop off the head. I learned how to hold a chicken upside down by the feet, with the wingtips in the same hand, and the chicken will usually dangle there with its head sticking out to be chopped. Some people drive two nails into the chopping block to make a v-shape, then sit the chicken's head between the nails. Pulling back gently will make the neck stretch out nice and long, so it is easier to chop.
Some people kill the chicken by cutting the throat, or cutting off the head, with a sharp knife. Some cut the head off with a pair of loppers (big scissors-like things for pruning trees.) Those methods are easier if the chicken is hanging upside down in some kind of cone, with the head sticking out the bottom. I've read of cones being made from a chicken feed bag (cut off one corner and hang the bag so that corner points down). I've read of kitty-litter jugs being used as cones. There are quite a few others made from various repurposed objects too.
Some people kill the chicken by the "broomstick" method (put broomstick on chicken's neck, stand on broomstick, pull on the chicken's feet until the neck comes apart.) There are more details that matter when you actually do it-- that description was just meant to give the basic idea.
I've heard of wringing the chicken's neck. I think it works about the same as the broomstick method, except that it is done with the hands and no broomstick. I've never tried it.
I'm not sure if you meant just killing the chicken, or also butchering it out to be eaten. For that, I use a sharp knife and a bowl (both borrowed from the kitchen.) For a surface to work on, it works to put a big pan on top of a sawhorse or table or garbage can. The "big pan" might be a metal cookie sheet, or a glass baking pan, or a steel frying pan, or anything else that can be cleaned and is large enough to lay a chicken on. Some people hang the chicken up instead of using a flat surface: tie a string from the chicken's feet to a tree branch, or to a nail in a wall.
I prefer to skin chickens instead of plucking, so that just uses a knife. I have also plucked the feathers after dipping the chicken in scalding water (fill a big pan with water, heat it on the stove, carry it carefully outside, dip the chicken.) I have also dry-plucked chickens (skip the scalding, just start pulling the feathers off.) People doing large numbers of chickens may benefit from a special scalding setup, and a plucking machine, but I have turned many chickens into dinner without those.
For removing the guts, I have only ever used a sharp knife plus my hand. The same knife is used to cut the chicken into pieces. The chicken, or the cut-up pieces, go into a bowl to be carried into the kitchen. They get rinsed in the kitchen sink, then put into a pan or bowl or plastic bag (depending on how long I want to store or it or how soon I want to cook it.)
The feathers and other parts I don't want to eat can go in a plastic bag and into the trash, or into a hole and buried (make it deep enough that predators do not dig it up again), or into a compost pile (again, watch out for predators), or to a dog or pig (a dog will probably leave the feathers but eat everything else you discard, depending on the size of the dog and how picky it is. A pig will probably eat everything else and the feathers too.)
At the end, I wash everything down and put things back where they go. Outside things (like a sawhorse) get hosed off, maybe scrubbed with a brush or a handful of weeds and rinsed again. Or if there's no hose, I carry a bucket or jug with water from the kitchen sink. The kitchen things get washed with soap and water just like normal dishwashing.
Basically, my tools are things I already have in my kitchen or garage, that all have other purposes. They go back to those other uses after I'm done killing (and possibly butchering) the chicken.