I'd like to consider this Azure feature:
https://www.youtube.com/watch?v=HxXs9_ggBfg
But it won't be of much use to me if I'm unable to configure a custom domain for it. I'm having no problem with custom domains for standard App Service websites, but this is for containers.
I've tried searching for documentation, but I'm coming up empty.
--EDIT--
The technology I'm interested in is Web Apps for Containers; thanks to @sam-cogan for pointing this out to me.
I believe the video is talking about Azure Web Apps for Containers (I would really recommend not asking people to watch a 30-minute video to understand your question). Assuming that is the case, then adding a custom domain name is the same as setting one up for a normal app service, you can do so through the custom domain name settings in the portal. The process is documented here.
If you're looking to use a different service in Azure for hosting containers then please state which one and I will update. All of these can have custom domain names.