I can't see how to do it in Eclipse.
http://developer.android.com/tools/publishing/app-signing.html#releasecompile
Then do I just copy back this stored away apk over the one in my bin directory?
No. You distribute it. Upload it to the Play Store. Or ship it on your Web site. Or copy it to ten million USB thumb drives and hand-deliver them to potential users. Whatever floats your boat.