As others have said, a simple right click in a regular Finder window will make it possible to "show package contents". 

For a file open dialog, use the "Go to location" functionality (<kbd>⌘</kbd>+<kbd>Shift</kbd>+<kbd>G</kbd>) and type the full path to a subfolder or subfile of the package. For example...

    /Applications/Skype.app/Contents

After doing that, you can browse around inside the folder normally.