If your stock turns to jelly in the fridge, it means you did it right!
Simmering the bones breaks down the collagen and turns it into gelatin; that's the very essence of stock-making. The gelatin is exactly what you want from the stock; at low temperatures it has a very jelly-like consistency, but at higher temperatures it melts and provides a very rich texture.
The more you reduce the stock, the more gelatinous it will become, so if it turned out stiffer than you expected, it's probably because you let a lot of water evaporate. Typically, when making stock, you use a very low simmer; just a few bubbles per minute. Any more than that and your stock will reduce.
But reducing is perfectly OK, and many cooks will reduce an entire pot of stock down to a few tablespoons; it's like bullion, only without all the salt and preservatives. Heavily-reduced stock is called glace de viande and is often used to give a sauce or dish a little bit of extra kick, imparting a powerful meat flavour. But you probably didn't reduce it that much.
If you're finding it to be too concentrated or gelatinous, even at higher temperatures, you can feel free to dilute it. For soups or sauces you can probably dilute it 1 for 1, but taste as you go along just to make sure you're not watering it down too much.
When only the top layer of a stock solidifies in the refrigerator, it is probably fat, which you should skim off and discard (or reserve). The rest is your actual stock, and in your case it sounds delicious.
Also, since you mention vacuum-sealing - I prefer to use covered ice cube trays for freezing stock. That lets you portion it out much more easily than a big solid blob.
My guess would be that you boiled the soup at some point, possibly for an extended period of time.
If you bring it to a full boil the fat from the meat will emulsify and distribute itself through the liquid. This is the same stuff that foams to the top, the "scum" that a lot of recipes (usually ones that say bring to a boil, then simmer) tell you to skim off. I would venture to say that this distribution of the fat into the liquid is also why it tasted heavier.
When making stock or soups, the most I'll heat them is to just below boiling and keep them at that simmer for a little longer to make up for not bringing it to a boil. When making stock, I strain once through a chinois and again through a piece of cheesecloth in a chinois to help reduce the floating particulates, but this won't really solve the cloudiness problem (though it will help some).
You can also try putting it into the fridge overnight so that the fat comes to the top and solidifies, but I don't think this will solve the problem, only help reduce it.
If it's just the stock that's gone cloudy, this page has some suggestions on how to clarify it, but honestly unless you're presenting it to guests, I see no reason to even bother. I've made cloudy stock and clear stock before and except for a slight "heaviness" difference, the taste is generally about the same. The cloudy stock sometimes has a more oily mouth feel, but it's not a major difference - and I've found some people seem to prefer the soups or rices I make with a cloudy stock.
If you have a hard time controlling the temperature on your range-top, you can use an oven-safe stock pot and cook it in the oven at 180°F to keep it just below a boil.
One other trick, depending on what kind of soup you were making - if you were making (for example) chicken soup, consider trying to turn it into a cream of chicken soup to hide the cloudy broth :)
Best Answer
I'm looking for the online reference, but I remember reading in Cook's Illustrated that they were able to substitute a bit of gelatin to mimic the mouth-feel of homemade stock. I did find a beef stew recipe that used gelatin.
Based on how you described your recipe, I would say that the long cooking of chicken bones is indeed what's missing. You might get a better result if you roast your chicken breasts and vegetables before cooking them in the broth.