To open the back of a Fossil Am-4065 your best bet is to use a watch case opener like this
You can also use a pair of needle nose pliers, but you then run the risk of damaging your case back. I would advise against it. I've had to deal with quite a few watches that were damaged in such a way.
The opener has an assortment of small pins. Usually you will be able to find some that will best correspond to the size of the grooves on your watch. Then the case opener can be adjusted so that the pins are securely fitted to the grooves so that you can then turn the case back off.
You will need to hold your watch securely while turning the opener, or it may slip out of the grooves and damage your case back. This is not a difficult task, you just have to go slow and carefully.
After you do so you should be able to carefully pop the battery out. Do not use a small screwdriver or other metal tool to do so. It can damage your watch. You can usually use your fingernail. After you do so you can simply read the battery number off and find a replacement. Many grocery store/drug stores and Radio Shack carry an assortment of watch batteries. You can also go to a website like Battery Bob
and order the battery, but for one battery it may not be worth it.
Your watch will not have the same water resistance after you open it. Usually a watch like this will at least have it's case gasket lightly wiped with silicone grease before it's re-closed (or have the gasket/O-ring replaced and treated with grease). This should not be a problem if you don't swim or bathe with your watch on.