Willow Trees make the best Charcoal for use in pyrotechnics.
Also if you just mix in 75% Potassium Nitrate (KNO3), 15% Charcoal Powder, and 10% Sulfur, then you aren't going to have black powder. You would have what is refered to as green powder. Which is about 30-45% less powerful then black powder. In order to get "Black Powder" all the ingredents need to be ball milled for 2-4 hours (atleast 2-4 hours for small amounts under 1000grams, don't know about larger, as I never work with anything larger then 300 grams at a time). It is vital that you only use lead balls, otherwise you are likely to get a spark, and blow up your ball mill. Ball Milling infuses the charcoal with KNO3, and makes it much more effective.
I actully did hear on this history channel show, that people use to boil down cow urine, and somehow were left with pretty good Potassium Nitrate. The Charcoal is gotten my simply burning a dried williow, but as for the Sulfer, I do not know of a place to get/make it that was around back in those days.
As for starting a fire with water, there are several different chemical mixtures that will result in flaming if H20 is present in any amount. The only one that I have ever used, is made by mixing; 28 % Ammonium Nitrate, 3 % Ammonium Chloride, and 69 % Zinc Dust. Mix the powders together, pour it in a small pile and put a drop of water on it. In about a second, it will bubble and smoke and instantly burst into a blue-green flame. However, after all powders are mixed, it becomes very water sensitive, and should be mixed on the spot, and used right away.
I have no idea what Greek Fire was made out of, but I know that whatever it was, it must have has its own oxidizer, otherwise when smothered with water, it would go out (as there is no more O left for the fire to burn).
I also can't stress safety enough. While BlackPowder is dangerous, if you follow all safety rules, and don't leave the BP in containers that would allow moisture to get in, then the it is almost 100% stable, unlike some other LEs and all HEs.
EDIT:
http://pearl1.lanl.gov/periodic/elements/16.html explained to me how/where people got sulfur, although it wouldn't be something most people would be able to get...