In a large (4-quart) saucepan, add the broth. Cover and heat over medium/high heat until it comes to a boil.
Add the prepared barley, cooked chicken, carrots, celery, and tomato paste, and simmer until the vegetables are soft (about 10 minutes). Add salt and pepper - taste and adjust seasoning as needed.
Serve hot, with a side salad. For thicker soup, cool to room temperature and refrigerate over night. Reheat the following day over low heat before serving.
I always cook the barley separately and add it to the soup already cooked. However, if you want to cook the barley directly in the soup, be sure to increase the broth (or water) according to the barley package cooking instructions, since the barley will absorb it's cooking liquid during the process.
If you want to use hulled barley or pearled barley, you'll just need to account for their longer cooking times. TheKitchn website has a great how-to on cooking barley.