I want to add a field for private notes in the article item administrator. Does anyone know how I can accomplish this. See attached for illustration.

You can add an extrafield in the backend and hide it in your frontend.

I did try that (along with a CSS display:none), however the content that is entered in the field still shows in the mark-up (ctrl-u on Chrome. )

So...how do you "hide it" in the frontend...there is no setting that allows a developer to hide the content from displaying on the frontend. A hack, still shows the content in the mark-up.

People on the site buy placement (ie: an article/item), so I need a place to keep administrative information such as an Insertion Order number and other private customer information.

@Τ

You need to create a template override and then in the loop which is printing the extra fields check the extra field alias. If it's the alias you want to hide then don't print it. Of course this requires basic PHP knowledge.