I want to write a table view that lists expenses. At the top of the table view I’d like a navigation bar to show a little bar that visually demonstrates how much of the budget has been spent (see image). The lower portion of that navigation bar will be a custom UIView.
How do I modify the UINavigationBar to include this custom view? Is that even the right thing to do? Should I write the top portion of that image as a custom UIView complete with buttons or will that be rejected because I’m reimplementing a standard view?

You won’t be rejected if you build your own custom view.
You can set UINavigationItem’s titleView property as seen in the docs here
http://developer.apple.com/library/ios/#documentation/uikit/reference/UINavigationItem_Class/Reference/UINavigationItem.html
Personally, I would build this as a custom view and recreate your own back button functionality.