Ack! Need help re: cleaning dirty, gunky nails

During a routine grooming session last week, I noticed a bunch of dried black gunk on three of Buddy's front nails and nail beds, but we were finally able to clean most of it off with a toothbrush, some warm water, and a little elbow grease.  The skin around those three toes was slightly pink and puffy, but wasn't oozy or anything.  But I have to confess, I kinda lost track of time during all of the holiday craziness, and now the gunk is back and his toes seem tender to the touch again.

Does anyone have any suggestions re: how to clean his nails besides what we've already tried, and maybe that won't be quite so traumatic for him, especially since the skin seems a bit tender?  I've read that some people use baby wipes, but I wasn't sure if we needed to avoid certain brands or types because of any ingredients that might be toxic to cats.

Also, does anyone have any recommendations as to what we can do about that slight pink puffiness so it doesn't develop into an infection, or at least keep the skin more comfortable until it heals?

Thanks bunches!

UPDATE:   Everything is all peachy keen now.  Yay!

This time we added some Dawn non-concentrated dishwashing liquid to the bowl of warm water, and the gunk came off much better with the toothbrushes.  Then we tidied up a bit with some fragrance-free, hypoallergenic baby wipes.

To our surprise and delight, the nail beds and surrounding skin weren't as pink and puffy as before, which was our biggest concern since Buddy has some ongoing health issues.  Hopefully now that we know what works for us, we can stay on top of this.  

I'm thinking this is awfully strange and maybe there is some underlying issue with his paws that needs to be looked at. Maybe there is a fungus or something growing there   I'd keep a really keen eye on it and if it starts coming back, get him a Vet appointment before you clean it all off.

Well, as best as I can recall, this seemed to start when I changed the litter in his litter box.  We've been using the same brand and type of clumping litter forever, but the texture suddenly changed dramatically -- from little bitty gravel like stuff to more of a fine dust.  So I bought some more boxes of our usual brand, thinking maybe we just got a weird batch, and they have the same fine dust-like texture.

FWIW, our FIV- cat who lives in another part of the house is suddenly starting to have problems with the cat litter sticking to his nails and nail beds, and the two cats have absolutely no contact with each other.  But at the time, there was so much going on over here and both cats were already so stressed and upset re: all the disruptions to their usual routine and environment, I was reluctant to start trying different brands of litter on top of everything else.  Buddy and Oscar are very much creatures of habit, and the least little change can send them into a major tailspin.

But as you say, maybe there's something more going on.  Fortunately, our holiday company has left and the household has returned to its usual routine, so it will be a heck of a lot easier to keep close tabs on both of our babies and go from there.

Thanks much!