I found an interesting article that made me think back on my own bio on my website. I keep wondering how I should balance the personal vs the professional persona. This article had some good tips on how to craft my bio, and I think I translate some of it to my cover letters, too.
The Resume is Dead, The Bio is King
I don't know that I would completely replace a resume with a bio, however. Most traditional job postings still ask for your resume, and I think they would look at you askance if you handed them something else. This is a useful guide for more of a website bio, personal letter, or elevator pitch. Typically these things would come into play in less traditional job-seeking settings (which I am sure many creative professionals are familiar with). Still, hold on to that resume, I'm sure it will come in useful at some point.