Place the chicken bones and carcass in a large stockpot and cover with water.
Bring the water to a gentle simmer over medium heat, and cook for about 2 hours, skimming off foam and fat that rise to the surface.
Remove the bones and chicken from the broth, then strain the liquid through a fine mesh strainer into a clean pot, discarding solids.
Return the strained broth to the stove and bring to a gentle simmer.
Add the diced onion, sliced carrots, and chopped celery to the broth and cook for about 20 minutes, until vegetables are tender and fragrant.
Meanwhile, shred the cooked chicken using a fork, pulling apart the meat into bite-sized pieces.
Stir the shredded chicken into the simmering broth, and season with salt and pepper to taste.
Simmer for another 10 minutes, allowing flavors to meld and the chicken to reheat thoroughly.
Finish by stirring in chopped parsley, dill, and a squeeze of lemon juice for brightness.
Taste the broth and adjust seasoning with additional salt, pepper, or lemon if desired.
Serve the hot chicken soup in bowls, garnished with extra herbs if you like, and enjoy its comforting warmth.