You can see the Video.js Playlist UI plugin in action below. Look at the source of this page to see how to use it with your videos.
When using a custom class, the plugin looks for the first element that matches the given class and uses that as a container for the list.
Using this option means the default styles WILL NOT apply; so, you'll have to define your own. This may be desirable or not - depending on your implementation. This example has been left un-styled to demonstrate this fact.