There is no definitive recipe for chicken stock, what I put in it on one occasion will probably vary the next. I'll always include some onion, carrot and celery, but if I have trimmings from other vegetables, such as the coarse part of leeks, then they will go in too. I always try to include some bay leaves at the very least, but thyme, rosemary and parsley stalks are additions that feature regularly too.
You can use a leftover chicken carcass from a roast, or a fresh one. Using a carcass from a roast makes a brown stock, which is rich and full bodied. Uncooked chicken makes a white stock, which is ideal for lighter dishes. Chicken wings make very good stock too, their high proportion of bone is what you want for a stock. Add giblets if you have them, except for the liver, which will make the stock bitter.
Always start with cold water, and only add enough to cover the stock ingredients. Hugh F-W likens it to making tea, if you use one teabag in a large teapot you'll end up with a weak brew. It's tempting to use more water than you should- after all, you'll get more stock- but it won't have much flavour or be able to offer much by way of gelatinous texture, which is the point of making your own stock. Don't let the stock boil when you're making it. Boiling causes the fat to emulsify with the stock which you don't want. Some people will have you clarifying the stock with egg whites and crushed eggshells, I'm content to just skim as much unwanted stock grease as I can off the top.
Chicken bones- e.g. Leftover carcass from a roast chicken, or a fresh chicken carcass or chicken wings
Vegetables- onion, carrot and celery plus any extras, such as leeks, mushroom peelings.
Aromatics- On this occasion, I used a one teaspoon black peppercorns, 2 cloves of garlic, and some rosemary, thyme and parsley.
If you are staring with fresh chicken and want to make a brown stock, start by roasting the bones on a high heat in the oven until they are golden brown.
Put everything in a heavy bottomed saucepan, or a stockpot if you possess one. Cover with cold water, poking any rogue bones or flavourings that stick out back under the water- if they aren't covered, they won't be giving anything to the stock, and you want to extract all the flavour you can.
Slowly bring the water to a simmer, and then turn the heat down to the lowest setting so that the stock just emits a few gentle bubbles. Leave for 2 hours, at the very least, about 5would be perfect. Don't allow it to boil at any point.
Strain the stock twice, once with a colander to remove the large pieces, and then with a sieve, preferably lined with a (clean) piece of muslin to catch everything else. Allow to cool, and place in the fridge. The fat will solidify and settle on the top, and then you can remove it easily with some kitchen roll.
You can keep this in the fridge for about two days or freeze for a later date.