Everything in Grand Clock works the same as with previous versions of the Apple software. One thing about the new iPhone 5S is that more people may be setting their iPhone to be passcode (fingerprint) locked. In that case, the background chime will not sound when the iPhone is locked. That was also true in iOS 6.
The volume of the chimes when played in the background is still controlled by the volume setting in Settings > Sound > Ringer and Alerts. It is also still muted when the silent switch in the silent position.
You can now turn on the background chime to hear it play in the background while you are doing other things in the foreground! The chimes will be heard as loud as your "Ringer and Alerts" volume setting indicates, so make sure that you adjust that accordingly. That setting is found in the Settings App, under General > Sounds.
This update also addresses problems that caused the application to quit unexpectedly on the newest iPhone and iPad. The pendulum has also been enhanced in appearance. More updates are planned soon to add the option to silence the background chime at night, and to play other high quality chime melodies.