Put the chicken (carcass, whole or parts) in a big stock pot and cover with water. Break up the carcass a bit to fit the pan if need be.
Bring to a boil, then turn down to a bare simmer. You don't want to boil it but just have it slowly simmering. You'll likely have some gray scum-like stuff come to the surface in the first few minutes. Your broth will be clearer if you just skim this off with a spoon.
Add the vegetables and seasonings, partially cover, and simmer (with barely a bubble breaking the surface). You don't have to peel or neatly slice the vegetables. Just break them into chunks.
If you are using chicken on the bone or a whole chicken, the chicken meat will be cooked after an hour so and will be lovely for dishes calling for cooked chicken. Pull the chicken parts out and let cool a bit on something that can hold the drained juices. When you can handle it, pull the meat off the bones and put the bones back into the soup pot to simmer another few hours. Put the cooked chicken in the refrigerator or freezer for later use. If you're going to freeze it, it will stay moister if you have a little broth in with it.
After 2-3 hours, turn the heat off and let the broth cool a bit. You can pick out the largest pieces of bones and vegetables and throw them away, then pour the broth through a strainer lined with cheesecloth to get all the finer pieces out.
You can use the broth now but if you refrigerate it overnight the fat will have risen to the top and solidified so you can remove it.
The broth might be thick and jelly-like. This is good so don't worry! But it's okay if it's not. Keep the broth 5 days in the fridge or freeze it for soup later.