The way I do it, and have for about 40 years, is to get a #10 can or similar size and a cover for it (I usually use a small piece of plywood) that will cut off the air when I want it to...
cut your cotton into 3"-4" pieces. Light the cotton cloth with a lighter or match. Hold a square or two until you're nearly burning your fingers (the idea is to get the entire piece of cloth burning before you drop it into the bucket) I guess you could use a pair of pliers to hold the cloth longer. Drop the cloth into the bucket and cover the bucket with the plywood to put out the fire. Wait for everything to cool. Take out the charred cloth and repeat. I usually drop 4 or 5 pieces of cloth into the can before I cover it.
I make mine on the bbq pit outside, usually in the summer, because the fumes can be pretty strong, I wouldn't try it inside necessarily. I just use an old dog food that's been burnt out on the fire. Cut the cotton cloth, usually old t shirts, into 2 or 3 inch squares, drop them loosely into the can until full, not packed. Then for a lid I just double layer a piece of aluminum foil on top, poke a hole in the center with a tooth pick, icepick or nail. Next I place the can in the charcoal fire and watch for the smoke to pour out of the hole. After several minutes when it stops, remove the can, let it cool completely, then open. Char cloth should be completely black, but fairly together and flexible. If it crumbles to dust you it was over cooked. You might have to try it once or twice and you'll get the hang of it and see what works best for you.
Try this. Once it stops smoking, take it off the fire and plug the hole with a toothpick or something. Let cool. Sometimes air can get in this hole and continue to "COOK" the cloth. You DONT want that.
Here's a solid method from Les Stroud's book, "Survive!" He recommends a metal container with a punctured lid(I used an Altoids Sours container) and a piece of cloth (Linen and cotton fabrics are recommended. I used a washcloth. I'm not sure what fabrics it contained, but it worked.) Stuff the fabric in the can, put the punctured lid on, and place it in the fire or on coals, lid facing up. The contents inside will start to smoke, then eventually catch fire, which shoots out through the hole in the lid. Once the flame goes out, it is time to take the container out and let it cool down. This is what resulted:
The charred cloth took and held a spark incredibly easily. Blow on it and it increases in size. Just one strike from my firesteel was all it took to get it started.