Picked up my first muzzleloader a few days ago and I am curious about post shooting cleaning/oiling. I have read a lot that hot water and dish soap is pretty common to clean them with but wondering if anything else would work better as far as solvents? I plan to use powder pellets to start with. Also I am wondering about oiling it for storage. I have Birchwood Casey Syn Gun Oil and also Otis Tech Dry Lube, I mainly use these on my shotguns. Would either of these be appropriate for the ML? Open to all suggestions, thank you.

Hot soapy water to clean. As hot as you can stand it. Put the breech end in a bucket of water, use a bore brush, then a bore mop. The bore mop will create suction as you run the mop up and down the bore pulling the water through the length of the barrel. Rinse the same way with plain hot water. Use patches and a jag to dry the bore. I run a patch with bore butter to protect the bore during storage. I use a soft toothbrush with soap and water to clean the powder residue from the exterior. Dry then oil the exterior metal.

if you wear gloves and get it really hot it seems to dry its self. wipe down and lightly oil or bore butter. bblackpowder solvents are nasty, stink and end up all over. and do not clean as well if you ask me.

Do not use hot water to clean a ML. Use cool-tepid water. You don’t even need to add soap but a drop doesn’t hurt anything. After it’s clean or just light gray (actually graphite used to coat the powder) run a patch of Balistol down the bore. Wipe down and lightly oil the exterior. It’s best to not use petroleum based oils, with mineral oil products the exception. Avoid waxy products for lube. Use animal fats or Balistol for patch lube.

If it's a percussion gun you can put the hammer down over a piece of leather on the nipple to close it off and then fill the barrel up with water.  Let it stand upright for a few minutes and then pour it out, then you can repeat and cover the bore with your thumb and slosh it around and pour again.  Do this a few times and it will remove most of the fouling, then you can use patches soaked in water and maybe a bit of ballistol as once the water evaporates the ballistol will remain behind as an oil.  This is for real black powder, I don't know how to clean the substitutes.  Depending how much you shot you will want to take the lock off eventually and clean the back side.  

If it's a flintlock there will probably be fouling on the underside of the lock.  You can also pull the barrel and use the bore mop technique to pull water in and out of the barrel with the flash hole submerged in a bucket of water.