Open it in chrome and chose request desktop site?
Best way I've found is to sideload Opera, there's an apk on the official site (awesome!), in settings you can set it to tablet mode which gives you a fullscreen experience and change the user agent to desktop, youtube works great this way.
Edit: you'll need a keyboard/mouse/trackpad or something that you can use to navigate the app as a remote will be really bad, if you have a shield tv there's a root tool to re-enable the silver triangle trackpad so I guess you can use that.
What box do you have?
www.YouTube.com/tv ??