It almost seems foolish to post a recipe for chicken soup - it seems like something at once so basic there is no point - and so varied it's foolish to try. But here is a basic version which could be altered in so many ways ...
1 leftover roast chicken
thyme (lemon thyme is nice)
anything else you may find lying around that seems appropriate
Break up the chicken and put it in a large pot and cover with cold water - use no more than you need for the amount of chicken you have.
Add some carrots - these can be cut in just a few pieces for this first stage.
Add some chopped onion - also in large pieces.
Ditto for celery, pepper, thyme.
Bring it all to a boil, then lower heat and simmer for about 3 hours - you can add a bit more water if you deem it necessary - but don't dilute it any more than you need to or you'll end up with dishwater.
The soup needs to cook slow and long because you need to get enough of the bones to give up some gelatin so the soup will have some body to it - it should jell after being cooled, unlike commercial soup which employs thickeners.
After the soup has cooked, strain it into a large bowl to get rid of all the yucky overcooked vegetables, and pour the broth back in the pot. Bring the broth to a boil, and add new smaller cut carrots and celery, and the rice, and anything else may want to add. Pick over the chicken bones and add all the small pieces of meat back to the pot. Cook just until the vegetables are tender.
Salt! You may have noticed I didn't mention salt anywhere - or if you haven't you should've! I never add salt until the soup is finished and even then add a bit less than I think is needed. Salt is quite easy to add, even after being served but is very, very difficult to remove! If you add too much salt you're left with only two options - dilute the soup with water and end up with something inedible - or I've heard you could add potatoes and supposedly they overcome the excess salt in some mysterious manner.