I have an RWS 34N (Nickel plated) Air Rifle with a 3 x 9 variable Air Rifle Scope mounted in .177 cal, and have taken out many, many squirrels with it. Not sure how tough a chicken is but squirrels are pretty tough to kill with an air rifle.
Good point on checking on your local laws...
Assuming you can kill varmints that are ruining your property... I'd go with a side cocking or break barrel RWS or Beeman. .177 will be flattest shooting, but .20 or .22 will hit harder.
Best ammo...GAMO RAPTOR PELLETS! I've seen one of these kill a small hog with a head shot...unbelievable.
You probably don't need the scope for chickens but its a nice to have. You will need an AIR GUN scope...not one made for a conventional rimfire or centerfire rifle. AirGuns have different recoil impulse and will wreak a conventional scope.
As usual, you get what you pay for...
Have fun!