You will want to use a deep-cycle marine type battery if you want to be recharging it. How big is up to you, it doesn't affect the fence charge per se it just affects how long the battery lasts before it requires recharging. Can't really predict how long a given battery will last for you before needing recharging, because it depends so much on what your fence size/resistance/load is.
It needs to be covered to prevent premature deterioration of the plastics in sunlight, the charger itself often needs to be covered too for the same reason (but there are a few exceptions -- check mfr instructions with your particular charger). Most people would either mount the charger on a post or building, or stick it on top of a cinderblock. You can either build it a little tiny-doghouse-sized hut, or in the case of many chargers you can just upend a big bucket over it, secured against wind.
Good luck, have fun,
Pat