The whole premise of Kathy Griffin's show (at least the episodes I watched) was that she is only sort of famous and has to scrap to keep her name out there enough to keep making a living.
Don't get me wrong, there's only so long it's interesting to watch a show about the life of one person. It worked with Ozzy for awhile, but only because he's not all there and because the whole family was over the top.
She was funny on Seinfeld. I couldn't take too much of her though. A little bit goes a long way.