Welcome to the Ranch
I would suggest you don't want
an inner class, but four: LeftMoveListener UpMoveListener etc. That way you get rid of horrible things like checking the action command. Always call classes which implement Listeners XYZListener rather than InnerClass.
You need to add the Listeners to the buttons otherwise nothing will happen. You cannot add the listeners from inside your inner class.
leftButton.addActionListener(new LeftMoveListener());
It seems weird to have a frame move by itself; is that really what you are supposed to do?
I tried it and got Exceptions that I was dividing by 0. You will have to change those 0s to something else.
Once I sorted out those problems, the frame moved, but not necessarily in the direction I expected
We usually discuss GUIs in our GUIs forum so I shall move you there.
[edit]Change listener names to start with CapitalLetters[/edit]