How to ensure that the chicken meat is not over cooked or under cooked?
It is fairly atypical to cook large, bone-on pieces of chicken&mdash;including breasts&mdash;directly in soup.  After all, who wants to find bones in their soup?  (Making stock is another story....)  

Bone on chicken breasts difficult to cook evenly, due to their shape.  Some parts are thin, and will cook through rapidly, and other parts are thick and require a much longer time to cook.  This makes it easy to have overcooked and undercooked areas.  They are also a little unforgiving in soup if overcooked, as they tend to become tough or stringy in texture.

Normally, chicken would be pre-cooked via another method, and then cut up or shredded, and placed in the soup in the last few minutes in order to heat through for service. My favorite method of preparing chicken to use in soup or other recipes is to roast it, still skin on for flavor, but any method will do.  The skin can be removed prior to adding the chicken to the soup, if you desire.

Note that this method will not add the flavor and gelatin (for that rich texture) from the bones to your soup.  Instead, as cook chicken parts for other dishes, save the bones and other scraps in the freezer.  When you have several pounds (a couple kilograms), you can use them to make chicken stock.  Homemade chicken broth is a fantastic base for soup, and will help you achieve an excellent flavor.

If you absolutely want to cook chicken breast directly in your soup, I recommend that you de-bone it.  Cut it into bite sized pieces.  As one of the last finishing steps in preparing the soup, reduce the pot to a simmer rather than a full boil, and drop in the chicken pieces.   This will permit the chicken pieces to poach gently in the broth.  Your soup is ready for service. once they are cooked through&mdash;my guess is about 15 minutes, depending on the size of the pieces, but check them occasionally.