When I boil (slowly simmer, actually) a chicken, I always put onion, celery, parsley and garlic in it from the start. Basically, I'm doing the same thing as Alan, but cutting out one step.
I then strain the broth before I refrigerate it. Once it's chilled (the next day), if I'm not going to use it right away, I skim the fat off the top and freeze the jelled broth for later.
I have found that if I boil the strained broth until it is reduced by half, the flavor is superior to anything one can get in a can.
*Note to Alan...when we get old and stove up, we tend to compromise. Your method is excellent.
We get by with a little help from our friends