I do not blame you for wanting a T/C rifle. The Hawkins is a great rifle. In fact I know where there is a USED but in perfect shape flintlock Thompson Center Hawkins. The rifle has a peep sight and brass front site. Also the original barrel was replaced with a Green Mountain .50 caliber 1-70 twist rifle barrel. So this would be an excellent roundball shooting rifle. He's asking $325 TYD only because of the condition of the rifle. New this rifle will run you around $550.00 and his peep is another $60.00 so you do the math. Its a nice rifle. He showed me pictures of it, but I have one identical to it, so I have no need for another one. Although at that price I might be tempted.
Since you are going to shoot a Flintlock, then there is only 1 powder to consider. Black Powder. Goex 2f or some of the other brands (other then elephant.. I will never recommend that powder) out there. And then some 3f or 4f for the pan primer.
The bullet will depend on the rifle twist. 1-70 is a round ball barrel. Mine shoots two inch groups all day at 50 yards. He claims his does a little better, but he uses a tighter ball then I do. If the twist is 1-48 then it will shoot roundball, conicals, even sabots. If the barrel is 1-32 then it will shoot conicals or sabots. If the barrel is a 1-28 it will shoot sabots and conicals real well.
Besides Thompson Center rifles, Lyman makes a good rifle. My Trade Rifle does a real good job shooting ball and conicals. I also have a Traditions Woodsman flinter and it shoots roundball real well and also 240 grain Cheap Shot sabots.
I agree, the T/C Hawkens is an accurate rifle. The first BP rifle I owned was a T/C kit in percussion. I do not know how good they in a flintlock because I have never shot one. The Lyman is a good choice if you want an off the shelf rifle. I have a friend that has one and I do like it for what it is, that is being a good off the shelf rifle. If you have the skills, I would suggest buying a high quality kit with a good barrel choice like Colerain's round bottom rifling.
Cayugad is right about the powder too, only black powder will work in a flintlock. Swiss is the best choice, though more expensive, the next best is K1K powder, then Goex. Anything but Elephant!
I use Swiss 3Fg in all my rifles with the largest being a .62 caliber. My biggest load is 85 gr. (although, I usually load 75 gr.) and get great velocity, quicker ignition, excellent accuracy, and not much fouling. If you use 3Fg in a caliber larger than .50, reduce the charge by 15% over 2Fg. For the pan, you can use either 4Fg or 3Fg. If the lock is tuned properly and the touch hole is in the right spot, she'll fire every time! My rifle will fire even upside down.
Notes on loading a flintlock. To laod a flintlock properly, start by having a clean bore. Charge the bore with a selected charge, ram down the patched ball until you feel it seat, gve it another light ram or two. Mark your ram rod and always seat it to that depth. With a percussion, you want the charge to be compressed more, with a flintlock, you want the ball touching, but the charge only sligtly compresssed. Do not turn the rifle over and slap it the way you would in getting the powder closer to a cap in a percussion. The opposite is true with a flintlock. You do not want the powder all the way to the edge of the touch hole. Fill the pan 1/3 to 1/2 full. If the powder covers the touch hole, it will slow the ignition down. Keep a cotton cloth to wipe the pan and the end of the flint between shots. It may sound like a lot, but you will fall in love with the flinter once you learn it.