I just use a tarp to cover my outside dog runs, but you could use anything like plywood or that fiberglass roof stuff you see on patios. I find the tarp is light and easy to get up and down.
for the mud we are dealing with that lately { although not in the kennel run as its covered and pretty much protected from that} but the yard im bringing in hog fuel to make a walk way around the bottom of our stairs out to the kennel , but you could use pea gravel, cement, concrete slabs, you could also dig down a foot or 2 if you want to do the work and lay sand and then gravel ontop and then the hog fuel or what surface you want. { we did this for our riding ring for the horses and never had flooding issues or mucky grounds}