Start with a really sharp boning knife. (long thin blade) The same technique is used whether you start with a split breast or a whole bird. Place the narrow end of the breast or the cavity of the bird on the board facing you. The breast bone sticks straight up, perpendicular to the rib cage. Slice between the breasts through skin and down to the breast bone. Point your knife straight down and use the tip to scrape the meat away from the bone down to the rib cage. The last inch of breastbone is cartilage so be careful not to cut into it but use it as a guide to scrape the meat off. Now work the tip of the knife along the ribs to separate the meat. The fat end of the breast has the shoulder and wishbone. Feel around to locate it and carefully cut it out, cut the meat off the shoulder and you're done. Now pull the skin off and you should have a breast, the bones and skin all separate. (You can leave the skin on if you like but if you do you must make it crisp during cooking) Clean any bits of organ meats from the bones, they seem to darken the stock. Put the bones in the stock pot and put the pot in a 375º oven for about 60 minutes.
Put a few drops of oil in a small frying pan and put the chicken skins in it to slowly crisp up. Don't make it hot enough to spatter all over the stove. When the skin is crispy on both sides put on a paper towl to drain, feed to the dog as a treat.
When the stock pot is cooled enough to work with add cold water to cover the bones and onion. Onion skin gives the stock it's amber color so leave everything on the onion, just cut it in half and put it all in the pot. Cut a carrot and a stick of celery into large chunks, 1.5-2 inches long is fine and put in the pot. Drop in a bay leaf and turn on the heat. Quickly bring the stock to a boil then turn it down to a slight simmer, cover and let it cook for 45 minutes at least. I get best results at 2 hours. If you simmer it too fast you'll lose a lot of water so you can add some if you need to. When it's done enough take it off the burner and let it cool down and settle.
You need to strain the stock. If you have a large sieve you can strain the bones or just take them out to make it easier. When pouring the stock, reserve the last 1 to 3 tablespoons in the pot, it has all the sediment in it. You should have a fairly clear amber stock. If it's cloudy don't worry, it's just particles of chicken floating in the stock. Let it cool completely then put in the fridge. Overnight the fat will clump together on the surface, remove it with a large spoon and dispose of.
That's it, chicken stock is used to make soup, gravy, sauce, risotto and many other dishes.