I've been reading through Google's Chromecast developer documentation and I can't seem to find any developer docs that discuss how to develop customizations for the Chromecast home screen, such as HTML overlays, etc.
There are Chromecast apps available that do show stuff on the Chromecast home screen, such as Dashboard Cast, https://play.google.com/store/apps/details?id=com.rir.dashboardcast&hl=en.
Does anyone know what approaches can be used with the Receiver API to allow this?
You cannot customize the home screen (backdrop); users can change what sort of feeds (images) can be used to be shown there. You, as a developer, do not have control over that and if it seems that some developers have customized that, in reality they have not; they are running an app there (like any other chromecast app).