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 element, the plugin uses the element that it was given as a container for the list.
Using this option means the default styles MAY NOT apply (it depends on whether the element has the "vjs-playlist" class) so, you might 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.