Good question, though I wouldn't limit recognition to just AKC registration (but I understand the point of the question as how are new breeds established and recognized by registries).
My understanding is that new breeds are developed by determining which traits are desired in the new breed and then selectively breeding for them. So, if I want a dog that can swim, track, and sprint, and with a specific body type suited for such tasks, I might mix a lab, bloodhound, and greyhound. Then, test the puppies to see which are best at the desired tasks, breed them, and continue to test and refine until I get consistent offspring. Of course, there is much more to it to ensure health, temperament, genetic diversity, etc.
I have read that there needs to be an established standard and the dogs should "breed true" which I think means that the offspring of new breed female and new breed male should be consistent with the new breed standard.
Kennel clubs, I believe, have their own criteria for recognizing breeds. For example, I've read that AKC requires an established breed club, standards, and a certain level of interest to recognize a breed. They don't recognize some breeds that are common in other areas of the world because there isn't a large enough presence in the US. The AKC has Miscellaneous
and Foundation Stock Service
groups for breeds seeking (I think) full recognition. This page
has some information on AKC recognition process.
But really, I just posted because I'm interested in the answers.