I don' t know a ton about Centerfire rifles, but I do happen to know Flintlocks pretty well........
Here' s how I do it........I was tought by Dave Ehrig " Mr. Flintlock" here in PA on the do' s and don' ts of cleaning a Flintlock.

First......I never ever touch anything petroluem based for cleaning anything having to do with the bore or ignition system (except the internal workings)

I use the T/C style " all natural" cleaning products.(there are others on the market) The goal is a " seasoning effect" in the bore, and if you use cleaning products that you would use on a centerfire rifle you are ruining your " Seasoning" .........well actually you' ll never even get one started.

The " Hot" water method of cleaning blackpowder fouling is,while good intentioned, not necessary and a myth when it comes to cleaning a blackpowder gun. Cold/cool water is fine and actually preffered. (Dave Ehrig actually cleans his barrels in a toilet!! Scrub away with a solvent soaked patch with the breech end in the toilet, and when you are done simply flush away the dirty water.....a couple flushes will have a nice clean look to it as all powder residue is removed)
You do not want a bone dry hot bore as it will again destroy the " Seasoning" effect by boiling off the natural protectants and opening the pores of the metal.
Seasoning a bore will promote less powder fouling, and easier loading between shots.
(While sighting in I do a light cleaning between every single shot......BUT NO OILS!)
If you Oil a bore with a petroleum based product you will not only destroy the seasoning, you will more than likely get an unpredictable first shot out of the oiled bore.

Here' s my cleaning process followed by what you will need (if you choose to do it my way)
CLEANING THE BORE
1. Remove the barrel from the stock.....
2. Remove the " Touch hole" (A good investement is an aftermarket slightly bored out version for more reliable ignition)
3. Place the removed Touch Hole into a small cup with just enough T/C " #13 Bore Cleaner" to cover the entire thing.
4. Over the Sink squirt some more #13 Bore Cleaner directly into the bore and let it sit for a minute or 2 while you fill the sink with cool water...........Place the breech end in the water (use a dish towel to on the bottom of the sink to prevent any scratching.........and don' t use one of Mom' s good ones!)
now once you have enough water in the sink to completely cover the threads where your touch hole was Take a #13 soaked patch on a jag and run it down the bore. Draw it back slowly in a pumping motion to draw the cool water into the bore. Be careful when you get near the muzzle, or you' ll cover the kitchen in nasty water! (Yes it will stink like sulfur....that' s the fun part......all sorts of chances for Mom to get mad at you[:@])
5. Run enough solvent soaked patches thru the bore in this manner until you start getting very clean wet patches. (You may need to use fresh water a few times in the sink depending upon just how fouled up the bore is)
6. Once you are satisfied that the patches are clean enough get the barrel out of the water.
7. With a #13 solvent moistened patch or pipe cleaner make sure the threads are free of fouling where the touch hole will be re-inserted. Also clean any powder residue off the outside of the breech area caused by the eruption of the flash pan powder.
8. Now run Dry patches thru the bore one patch for each pass until you have the bore completely dry.
9. The final step in cleaning and protecting the bore is done with a patch coated with T/C " Natural Lube 1000" Bore butter.........once down and once back and the bore is now clean,protected and on its way to a proper Seasoning.
10. Fish your Touch hole out of the cup full of solvent and run it under the tap.....DON' T DROP IT DOWN THE DRAIN! []
WIth a cleaning patch and pipe cleaner make sure your touch hole is clean and clear.
11. Apply a small amount of " Bore Butter" to the threads and re-install the Touch Hole
12. Finally wipe down the entire barrel with a Silicone or Oil Moistened rag and Set it aside. That part is done.

CLEANING THE IGNITION SYSTEM
1. Remove the Flint and leather carrier and set it aside for now.
2. Remove the entire lock from the stock.
3. With a #13 bore solvent moistened rag or cleaning patch ,wipe down everything external to free it of powder residue.
4. Turn your attention to the inner workings........make sure everything looks free of residue and " gunk" If its a mess you can flush it with a normal Bore solvent first , dry and then apply a light coat of spray on gun oil. Don' t get crazy with it tho.
5. Now take your flint and clean it with #13 bore solvent to remove all the powder residue. (I personally like premium Agate flints for all of my shooting and hunting........they throw a hotter bigger spark, are more reliable and you can see fouling better on them. They may wear your frizzen more than a softer English flint but I' d rather the gun actaully go off everytime I pull the trigger)
After you have the flint clean here' s a trick my buddy Frank came up with years ago and it has really enhanced the ignition of our guns.
Buy a small can or CRC ' Electronic Contact Cleaner' from your local Auto parts store........Spray the flint (on a piece of newspaper, and then wipe dry with a clean patch.....careful not to get your finger oils on the contact surface of the flint)
6. With the same CRC Contact cleaner lightly spray ONLY the face of your frizzen and wipe dry. Also wipe the flash pan bowl with CRC.
What the contact cleaner does is ensure a perfectly dry and oil free flint to metal contact that will consistently throw the hottest spark. NEVER EVER EVER wipe your frizzen or flash pan with any oil.
GREAT TRICK!
Reinstall your flint into the leather carrier and back into the hammer........center it carefully, and align the flint so that there is about a 1/16th of an inch gap between the flint edge and the face of the frizzen when at " Half cock"
7. Put the lock back into the gun, wipe down the external parts , EXCEPT FOR THE FRIZZEN FACE , FLASH PAN AND FLINT or LEATHER with a silicone cloth.
8. Give the flint and the frizzen face one last wipe with a CRC Contact cleaner moistened patch and your' re done.

Finally wipe the entire external parts (Lock Stock and Barrel)down again for good measure with a Silicone cloth ...again avoid the flint and frizzen face.

Sounds a little more time consuming than the other methods, but IMO its the right way to do it and the way I' ve always done mine. Its also the exact way I intend to keep my new Inline clean. No offense intended to the other guys who have offered their help, this is just my opinion on how a muzzleloader should be cared for.
If you are careful and keep everything clean and dry misfires should not happen.........I can honestly say that in all the years I have hunted with a flintlock I have NEVER had a misfire while hunting, and lock times are terrific. No hang fires........just a KABOOM!

Here' s what you' ll need to duplicate my particular cleaning process:

1. T/C #13 bore cleaner
2. T/C " Natural Lube 1000" Bore Butter (tube)
3. Cleaning patches 2-2.5"
4. Cleaning Jag.(in your caliber)
5. Patch puller (in case you have one slip off the jag)
6. CRC " Electronic Contact Cleaner"
7. Cool water, a sink(or toilet) and an understanding Mother.
8. A good gun oil like Shooter' s Choice etc. for the internal lock
9. Silicone gun Cloth
10. A barrel pin remover (you can use a hammer but its crude)
ADDITIONAL GOODIES:
11. A good initial investment is also the Agate flints, and a bore out touch hole.......these will make shooting much less frustrating right off the bat and will very much enhance the reliablity of your gun to fire in hunting situations.
12. A good storage container such as the Plano style vertical shooting boxes makes hauling all of your supplies much easier
13. You might also want to get yourself (if you don' t already have one)a gun cleaning mat such as those green ones manufatcured by Remington. Or a Cleaning station that holds the gun with compartments underneath.
14. A small battery powered bore light that you drop down the bore is a nice cheap tool to have as well to confirm your cleaning the bore well.

Hope this helps.........have fun with it, these guns are a ton of fun to shoot, and they are more than capable hunting arms.
If you have any other questions please feel free to e-mail me or ask here.