We need to stop condoning the killing of cats on this site. There are other options!
Directly from the ASPCA :
Does Eradication Work?
Eradication, the deliberate and systematic destruction of a feral cat colony, by whatever method, almost always leads to the vacuum effecteither new cats flock to the vacated area to exploit whatever food source attracted the original inhabitants, or survivors breed and their descendants are more cautious around threats. Simply put, eradication is only a temporary fix that sacrifices animals' lives unnecessarily, yet yields no psitive or beneficial return.
What Is Relocation and Why Doesn't it Work?
Many communities have rounded up colonies of feral cats for either euthanasia or to relocate them to another area. This never works. Feral cats are very connected with their territory. They are familiar with the food sources, where to find shelter, resident wildlife, other cats in the area and potential threats to their safetyall things that help them survive. Relocation of feral cat colonies is difficult to orchestrate and not 100-percent successful even if done correctly. It is also usually impossible to catch all of the cats, and it only takes one male and one female to begin reproducing the colony, Oldham states. Even when rounding up is diligently performed and all ferals are removed, new cats will soon move in and set up camp.
http://www.aspca.org/adoption/feral-cats-faq.aspx#eradication
Chicken coops should not be designed to keep your chickens in, but to keep everything else out. I would suggest looking into changing the door latch.