Firstly, if cooking on coal or flame, first coat the OUTSIDE of your non-cast iron pots and skillets with the dish soap. Clean up will be much easier as it's that layer of soap (and not the pot) the soot adheres to. Every Boy Scout knows this trick, or at least he should.
Secondly, if there is something else stubborn you must absolutely remove, make a paste of baking soda and water. Cover the spot with the backing soda and water paste, leave it on for twenty minutes or so, then scrub and rinse.
Baking soda, water, and vinegar creates carbonic acid which breaks down making "scrubbing bubbles" --- good for lodged drains, sieves, and such.
Baking soda and vinegar are really the only household scrubbing cleansers you'll truly need for most applications. Sometimes you use them alone. Sometimes combined. Depends on application/use. Saves a ton of money too.