I've found that cold water coffee brings out the best in coffee, at least for my taste, as the hot water used for most coffee brewing dissolves some of the bitter oils in the beans, so this time consuming but gentle method can make cheap coffee taste as good as a more expensive coffee.
My method uses a gallon jar and a fine mesh sieve that fits inside it to make this coffee. Amazon sells them, but there are other choices that probably work just as well.
The cheap coffee I use is a very fine grind. Most recipes for cold water coffee recommend that you use a coarse grind, as the fine mesh sieve that comes with the brewer isn't all that fine, and quite a lot of tiny coffee grounds leak into your final product, and the coarser grind minimizes this. That said, coffee grounds do not enhance the final taste of the coffee, which is why I recommend that when you stir the brewing coffee, you do it gently.
What I do is to put about a gallon of filtered water (Brita water works fine, you could even get away with tap water if your municipal water is good) and then I add two and a half cups of Costco's cheapest fine grind coffee into the central sieve of the maker. I very gently stir the floating coffee grounds (that sieve is delicate) with a plastic rod (chopstick, spoon handle, whatever's around) until most of them are waterlogged, put on the top of the jar, and let it sit for a day.
The next day, I remove the top, give it a bit of a gentle stir again, and replace the top.
The day after that, I again remove the top, give another gentle stir, and replace the top.
The day after that, I remove the filter with its coffee grounds and let it drip into a coffee cup. I take a wide mouth funnel (I use the cut off top of a 2 liter soda bottle) and pour the coffee into a gallon container (I use an old, well washed, gallon milk jug). The majority of the coffee will not have much in the way of tiny grounds in it. After I decant all but a cup of this into the container, I stop. Then I take a coffee filter (the ones that fit the Mr. Coffee brewer, about 6" (15cm) in diameter) and put it into another 2 liter soda bottle with the bottom cut off. This is then put into the opening in the gallon container. I then pour the rest of the coffee into the filter. The filter will catch those muddy grounds, but the rest of the coffee will take upwards of 10 minutes to drain. After the level of the coffee is down enough, I take that coffee cup that the grounds have been dripping into and pour this in with the rest of the coffee. It might take another hour for all of the coffee to drip into the container, as there is now a thick layer of very fine coffee grounds clogging the filter.
Finally, I add water to bring the amount of coffee to a gallon, cap it, and put it into the refrigerator, ready for use.
My Alaskan friend Mike points out that the paper filter itself can be very carefully rinsed, dried, and reused.
When it's time for coffee, pour yourself a cup, add what sweeteners and dairy you want, and microwave it. My microwave takes 10-12 ounces of coffee to almost boiling in 2.5 - 3 minutes. Enjoy!
Another trick you might want to use is to keep the opened coffee as fresh as possible by keeping it in the freezer in a tightly sealed container.
But wait! I don't want to wait four days for my morning Joe!
Simple. Keep 2 gallons of coffee in the 'fridge. As long as you don't go through a gallon of coffee in four days, no problem.