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 passcode 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 is in the silent position.