How do I optimize my chicken stock for the best flavor at the lowest cost?

Chicken stock (or any stock) is made from bones; that generally means backs, necks, and carcasses - which are generally the cheapest "cuts" because they have almost no meat and aren't useful for much else. Sometimes, if you approach a butcher with high turnaround, you can get them at a steep discount, well under $0.50/kg, because they just end up throwing most of it out.

If you add any meat whatsoever then you are essentially making a broth, not a stock. I repeat, stock is just bones, simmered long enough to denature all or most of the collagen, usually no less than 8 hours.

I find that most people (myself included) prefer brown stocks, so roast them first. You can sauté, but it's not as effective as a slower roast because the heat doesn't really penetrate far inside the bones. It also helps to cleave the bones before simmering (after roasting) to expose the marrow; it makes for easier extraction, although some people don't like the marrow flavour as much.

The least expensive way to add more flavour is probably to add the dry stock mix or bouillon cubes filled with salt, MSG, and other flavour enhancers. Of course it won't really taste like a homemade stock if you do that. If you want a good stock then the traditional way to flavour it, which is also very inexpensive, is to use a mirepoix, which is just a 2:1:1 ratio of chopped onions, carrots, and celery. You add this about 1-2 hours before the stock is done and remove it before clarifying.

Going up the cost/taste scale is a bouquet garni, which, although the herbs that go into it can get fairly pricey, the amount you use is minuscule, so it costs practically nothing if you have other uses for the remainder of the herbs.

Don't even think about trying to put "expensive cuts" of meat into a stock to improve its flavour. You'll just be wasting the meat, which loses most of its flavour when boiled/simmered, and turning your stock into an unsalted broth, which makes it less versatile and generally perceived as blander.

Now if you want to make a soup with better cuts of meat, that's great; in that case, the texture matters, and I'll occasionally poach some boneless thigh or breast meat to use in a chicken soup (made from the aforementioned stock) for the sake of convenience. But I emphasize that this is done after the stock is made. The soup will also have other ingredients added such as chopped vegetables and of course plenty of salt.

Stock isn't supposed to have an exceptionally complex flavour, that's why it's called "stock" - it forms the base of other dishes such as soups, broths, or anything that calls for flavourful liquid (like a risotto). Focus on making a good clear stock from just bones and mirepoix; it gives the best result because you tweak it later according to how you use it, and it costs you almost nothing besides the energy it takes to keep the pot heated.

Aaronut's answer is great. I have one thing to add for the budget-minded chicken broth maker. Costco now carries a 1lb (or close to it) jar of Better than Bouillon, the organic, reduced sodium variety. This product did very well in America's Test Kitchen taste test, about equivalent to Swanson's canned broth. It (like Swanson's) is pretty insipid by itself, but it works great to stretch or enrich homemade broth. I can't remember exactly what Costco charges, but it's less than the small jar at the supermarket.