Automatically post YouTube videos from a certain channel?

If I’m understanding your question correctly, you want YouTube links to be published on discourse in a private category when a video is posted. That would be the same system as it links to a category itself, pretty much the same thing. Install the RSS plug-in that was linked above, then use that YouTube RSS generator and plug and apply it in. Follow the other instructions on the RSS plug-in topic to continue to set up in specify the category you want it in.