You need to find the bpy.data.images entry for the image. If you know the name of the image but not the path you can use bpy.data.images[NAME]
Once you know the entry, the full file path is in the filepath member, so something like bpy.data.images['foo.png'].filepath will return the full path to the image.
If you don't know the name, but you know you have only a single camera background image, you can use bpy.data.cameras['Camera'].background_images.items() to fetch a list of all of the background images associated with the camera. Here's what the list looks like when there is only one image:
[(0, bpy.data.cameras['Camera']...CameraBackgroundImage)]
You can find the bpy.data.Images entry from the second member of the tuple, so
items = bpy.data.cameras['Camera'].background_images.items()
camera_image = items[0][1]
filepath = camera_image.image.filepath
Will give you the full path to the first camera background image. For an non-camera background image you need to find the empty associated with the background image. Here's how to get the filepath for a background that's associated with an empty with the default name
filepath = bpy.data.objects['Empty'].data.filepath
UPDATE: about relative paths
If you have blender set up to use relative paths, filepath will be a path that starts '//' indicating that it is relative to the directory where your blend file is stored. To pass this as an absolute path to functions that require such things, instead of filepath, use bpy.path.abspath(filepath).