Cut the chicken into smaller pieces so they better fit into the pot and they break down easier in the broth. I typically remove the legs and wings. I then separate the back from the breasts. Then add the chicken pieces into the pot.
Cut the carrots and celery into 3 inch pieces. Peal the onion and cut it into fourths.
Add the cut vegetables to the same pot as the chicken. Then add enough cold water to cover the chicken bones and vegetables by an inch or two. Cold water it important as it prevents your broth from becoming cloudy.
Add the thyme, parsley, garlic, bay leaves, peppercorns, and salt to the pot and give it a good stir.
Add the pot to the stove and bring to a boil on high heat to bring it to a simmer as fast as possible. The longer it takes to come to a simmer, the cloudier the broth will be.
Just before the broth comes to a boil, reduce the heat to low and simmer for two hours. Skim all the white scum that floats to the top of the broth so it doesn't make the broth cloudy. This is an important step that needs to be repeated as much as needed (I usually check on it every 30 minutes or so).
Once the broth is done simmering, remove the large pieces of chicken, bone and vegetables. Make sure to keep the chicken for a soup or other dish. Discard the bones and remaining vegetables. Then strain the broth into a large bowl and chill in the fridge overnight.
The next morning, skim the fat off the top of the broth. This helps to keep the broth clear. Once the broth is skimmed, you can either use it right away, store it in the fridge for 3-4 days, or freeze for up to six months.