It's really not that important. If time permits I'll put them in a tank for a few days and it will clean them out a bit. If it wasn't possible to do, I would not hessitate to skin it the day I caught it. I'd just spray it with a garden hose to wash off most of the mud. You really do not want to get it on the meat while you are skinning it. I would without a doupt not loose any sleep over skinning one the day I catch it.
To kill it, take a board and run a nail through it at an angle. The nail has to be long enough to pass through the board a few inchs on the pointy end. Slide the nail under it's bottom jaw and pull him back dragging his head out and chop it with a hatchet. That's the fastest easiest way I've found to kill them. You can also hold it up by the tail, and it will pop his head out to try and look up then you can cut it with a hatchet. It's a little harder that way though. Also be careful with the severed head, it could still bite down.
If you do decide to keep it, be aware that they are master escape artist. They will get out of places you would never expect.
Snapping turtle is one of my favorite wild game meals down here in the bayou's.
We always stored them in an old washtub for a few days, replacing the water daily. I don't ever remember going more than a few days, never more than a week. Alot of times we hosed them off first. As mentioned, you need a lid on whatever you store them in. We always gave them a wack in the head with a hammer and then drove a large screw driver threw the bottom of its head and then nailed into an ash tree to clean it. First thing we did was cut off the claws.