Save in a bag in your freezer along with all chicken bones you come across. Cleave through the bones to expose some collagen before freezing (consider roasting all ingredients before hand). Add whatever vegetable bits you don't use in your cooking - including things you wouldn't eat (but nothing poisonous) to your bag, especially onions (including skin and root), celery, and carrots.
When you have a good amount, put in a pot and just cover with cold water. A bit of vinegar helps promote a nice gelling of your broth. Tomato paste, peppercorns, and a bay leaf will add to the flavor. Simmer gently for around five hours. Remove. If you had fat still on your meat you may want to chill until the fat rises to the surface to remove the fat.
Your broth will keep 3 to 4 days in the fridge or it can be frozen until you have a recipe which calls for chicken stock or broth. Your homemade broth will be an excellent addition to sauces, a great base for soups, or an excellent liquid for cooking rice, couscous, or quinoa.
It's down to what the chicken ate while it was alive. Saturated fat sets, olive oil sets if you chill it, but not otherwise, and a number of seed oils do not set (rapeseed for instance). When you make a stock which has solidified fat on top, that's saturated fat, so I'd hazard a guess that the stock where the fat doesn't set means a healthier eating chicken, because it contained less saturated fat to start with.
UPDATE: Thank you to the person who bothered to do the research and said my answer 'might have some merit'. Chickens are no different from human beings - the fats you put in are the fats floating round your bloodstream and depositing in various places; think about corn fed chickens, where the fat composition is slightly different, not to mention the colour of the flesh itself. That will be a partial explanation; when taken together with the fact that not all chickens, even in the same flock or brood, get to eat the same diet, because the pecking order dictates that some free range birds don't always get the pick of the food, explains differences in chickens from the same supplier. Of course, if you can come up with another explanation, I'd be delighted to hear it...
UPDATE 2: Perhaps I should have been clearer. I am not for a moment suggesting that the fats eaten are deposited in their original form, but if you know anything about biology (chickens or otherwise) then you'll know that certain synergies occur, depending what's put in, which change the composition of any fats deposited within the body system. Hence the connection between eating lots of saturated fat and having high cholesterol in humans, for example.
UPDATE 3: Rumtscho: Can't find any scientific evidence so far to prove this theory regarding chickens, but, for interest's sake, and to prove how much of a difference it can make, farmed salmon in Britain no longer has a balanced omega 3/6/9 ratio, as it should do, and still does in the wild. It's because the feed had to be changed, and the consequence of that has been a much higher level of omega 6 in particular. I'm still looking for something on chicken.
UPDATE 4: Now I've had time to look properly, it's not at all difficult to find scientific evidence, there's plenty of it. There's a study carried out by The American Society for Nutritional Science in 2000 comparing the fat deposition (and other metabolic processes) between chickens fed the same diet, but one lot with saturated fat included in the form of tallow, and the other lot with polyunsaturated fats. The fat deposition in the birds fed tallow was greater, and the composition of the fat contained more saturates compared to the polyunsaturated group. These results reflect previous studies (Sanz et al 1999 and 2000).
Effectively, it's like everything else - you get back what you put in.
Best Answer
If you are cooking the chickens whole, and I think you are from your description, then your cooking liquid has not had sufficient contact with the bones to extract the available gelatin. You should absolutely be able to make a stock with the carcass later.
As you say, even with the nice gelatinous cooking liquid you have from your first step, you don't have a whole stock without the aromatics. My advice is to make your stock with the carcass as usual, but add your cooking liquid too for a nice boost.