You would have to adjust the neck tension, adjust the saddles and re-intonate the bass. I do not think you would have to do anything with the nut. You can find a few 5 string sets with C, and there is a GHS set that's made for this. There are a couple of other companies that make such a set, but you can go to
http://www.juststrings.com/bassguitarsinglestrings.html and likely find a C to go with your favorite 4 string set.
An EADGC set would allow you to make chords and double stops higher up than the normal tuning, which tends to sound particularly good on a fretless.
I have considered doing this on my 5 string fretless and just may do it.
