If you use inner scaling, you won't be able to fit every screen perfectly. It's literally in the name, it will scale (i.e. change size) to fit the screen. If your background is tiled and repeating however, you're in luck, because you shouldn't need to come up with a custom image for your background. Just slap that puppy into a Tiled Background.
If you have stuff you need to always need to be in certain corners or positions of the screen, look into the Anchor behavior.