A large cage trap and then shotgun and then shot gun to dispatch. Shotgun certificates are straight forward to get, I got mine 2 years ago again at the time had no land was a member of no clubs etc, you do not need to given a reason for wanting one, however you may be asked why you want one, dispatching of trapped fox thats been eating my chickens seems resonable. You will need secure storage, usually a purpose built cabinet with 2 locks is sufficent. If you want to shoot the fox a would recommend a rifle, this are harder to get, your 5 acres may be approved dependent on the lye of the land hill etc, the bullet needs to hit something if you miss, but a FAC which is required for a rifle is a little bit more complex to get, however if you don't have a criminal record or a history of mentel health problems, you should be able to get one just might have to jump a few hoops.
With trapping get some meat in but don't set the trap so the fox gets use for 3-4 days then set should get the fox, other trapping tips, bigger is better and use gloves to avoid the human smell. You will never stop foxes completely, so keeping stock in pens does keep them safe, but also keeping numbers down will help.