After spending so much time trying to get this to work myself, I felt that I had to write this post to help clarify much of the mystery behind Automatic Background Updates in WordPress for others. Hopefully it helps someone else out there.
This post, Part 1, will cover the basics of having your plugin automatically detect a new version when your plugin is not hosted within the wordpress.org repository. In Part 2, I will discuss the methods you can use to have WordPress actually upgrade your plugin for you now that it has detected that there is a new version to be installed.