Using your bathtub is more than a bit extreme. Not only is it unnecessary to clean your gun in the bath tub, but it will take longer to clean the bathtub than it took to clean the gun. Personally I canlt take another lecture from my wife questioning my logic with respect to doing something so unnecessary. (okay, stupid).
The original cleaner for black powder is simply water. I think you have a T/C Hawken which has a take down breech so you can remove the barrel from the stock by pulling out the wedge and lifting the barrel from the stock. All you need is a bucket of water (I use a large plastic coffee can) to put the breech in after you remove the nipple. A damp patch will force a jet of water through the barrel, flash channel and nipple seat with more than enough force to remove the carbon from the bore. A couple of drops of oil dissolving dish soap can be added if you use an oil or grease based lube. Of course you can use Moose Milk, it works too as well as WD40, Brake Cleaner, Winter formula Windshield Washer Fluid, Windex or the Generics. There are many other concoctions that clean also. Cleaning a muzzle Loader that uses Black Powder and most of the substitutes is easy since so many solvents dissolve the fouling. Once the bore, the breech area, nipple seat, lock including the hammer you want to dry the bore and get rid of the water. Now is the time for a WD40 or rubbing alcohol to displace any remaining water. A blast of compressed air is good too, but a wet WD40 of alcohol patch works. I like to run a patch with Ballistol down the barrel to introduce some oil to the barrel and see if any fouling remains. Don't always expect a white cleaning patch, gray is good. Finally a rust inhibiting lubricant to wet the patch ant run that patch in the bore, over the barrel and lock. Store the gun with the muzzle down to allow any accumulated oils to run out and for the rust inhibiting film to dry some. After a couple of days run another patch dampened with Ballistol or Barricade and the gun should be good for long term storage.