I did something like this recently with Skype and Ubuntu 13.10. You can set up auto-login on Ubuntu, auto-start on Skype, and set up Skype to auto-answer and auto-activate the camera. In theory, it could get torpedoed by some kind of random dialog box, but it hasn't happened in two months. To make a it a little more bulletproof, you could set up a cron job that reboots the system nightly.
One warning: since Skype was acquired by Microsoft, the standard Skype login dialog was replaced with some kind of weird integration to Microsoft's outlook.com single-sign-on. When Microsoft is having system problems (like they were a couple months ago), it appears to lose all of the single-sign-on tokens, which will drop the system back to the login page, and you will be stuck. However, this only happened once in the past couple months. (Yes, things worked far better before Microsoft bought Skype, but it's not a lost cause.)