1. Put the chicken and the water into a big pot over medium heat. Bring slowly ALMOST to a boil. Skim the foam that will rise - you'll have to do this 3-4 times until it stops forming. This is the blood and other impurities being heated out of the chicken flesh.
2. Now add the vegetables and seasonings. If you add them in the beginning, they will get in the way of skimming. Some people add leeks, too, but I find the cleaning of them too tedious. I don't add much salt to stock, since I don't know just what it will be used for later. Cover and let simmer on low. Don't let it come to a full boil. How long should it simmer? At least 2 hours, if you need it "quickly." But you could let this go all day and just strain it when you are ready to go to bed. It perfumes the house nicely.
3. Everything will have fallen apart in the pot by the end.. I use a wire skimmer to remove the big chunks, then pour the rest through a fine-mesh strainer into a smaller pot to cool. If you need the stock to be even more clear of particles, lay a few layers of cheesecloth across the strainer.
I cover the stock and allow it to cool for a little while, then refrigerate it overnight. The fat hardens on the top and is easier to remove. The stock gels in the fridge - this normal and my kids call it "Chicken Jello." If I am leaving it in the fridge for use in the coming week, I don't skim the fat until I use the stock. If I am freezing it, I remove most of the fat. Then I measure it into labeled storage containers. I would keep it for a week in the fridge, or 3 months in the freezer - although I make stock at least once a month, so mine doesn't last that long. Remember to label the containers - frozen chicken stock looks just like pork or fish stock.