
Mostly your breed club or one of them if you have more than one will put your name on the judging list after you have shown for a while and shown interest in the breed by owning and going to breed seminars and taking and passing the exams set at the breed seminars. You also need to show enough interest in showing and judging to take the 3 exams set by the KC and pass them also a good idea to start stewarding particularly for the breed you are interested in.
Once you have done the above you may be asked or you may have to ask to go on the breeds C list, then you wait to be offered classes. When you have started judging you will acquire numbers of dog you have assessed and will eventually once you have judged the number required by the Breed Club be moved up to the B list and then to the A3 list. Once you are on the A3 list you will have to fill in a form for the KC, a sort of CV and they will either give or refuse you permission to give CCs at Championship shows, if you get permission you will be assessed by a senior judge within the breed who will either pass you or not, please note the judge that assess you is not going to question your opinion or placing of the dogs but the way you handle the ring and your exhibits.
Different breeds and breed clubs have different requirement before putting you on their lists but these requirements are available from the secretary of the club and many list them on their web sites. The 3 KC exams are about the construction of dogs, the handling of dogs and how to run your ring when judging. When I took them one was a written paper, and the other two are assessments of your performance.
Hope that helps, and good for you to be interested go on and have a go.