Hello cleanqueen,
The type and quality of the paint greatly affects how you clean a wall and how easily dirt comes off. Below are a couple of paint types most often found on interior walls and a little information for you.
Enamel PaintsA very common household paint, especially for kitchen and bathroom walls, this surface is stain-resistant and can handle a moderate scrubbing. Use a neutral cleaning solution with a small amount of detergent and a white, nylon-backed scrub sponge. Only use heavy-duty cleaners or abrasive cleansers when you're willing to take your chances on ruining the paint and possibly having to repaint the area.
If you have latex enamel paints, don’t let them remain wet for more than a minute. Oil-based enamels are more water-resistant. Keep in mind that gloss enamels are the most durable and washable, followed by semi-glosses and then satin finishes.
Latex Flat PaintAnother common household paint, flat latex is not as washable as enamels. Heavy-duty cleaners or hard scrubbing can remove the paint along with any dirt. Use mild detergents and gentle scrubbing, and don't let any solution sit on the surface for more than a minute.
Cleaning Methods for Painted WallsHere are some cleaning methods you may have some luck with. Keep in mind you may end up having to repaint your wall. If you do end up having to re-paint, you will still need to get some of the grease up or sand it off so it doesn't bleed through the new paint.
Cream of Tartar PoulticeMake a thick poultice of cream of tartar and water, you only need to use just enough water so it will stick to the wall without drying out and falling off the wall.
Place a piece of plastic wrap over the mixture and let sit for a while then wash it off with warm water.
Cream of Tartar will suck out the oil as it dries. If this is working for you, you can repeat the process.
GoopGoop is a hand cleaner made for removing grease. Rub this directly over the affected area then wash off with warm water.
TSP (Trisodium Phosphate)I know you stated you don’t want to use chemicals, but this is the Gold Standard.
TSP is a heavy duty cleaner which neutralizes and floats away grease, grime, dirt, and oil. TSP contains no phosphates, soap, lye or abrasives and is a strong and effective cleaner.
Do test in an inconspicuous area first and follow the manufacturer’s instructions as it can cause paint damage if it is mixed too strongly.
Be sure to wear gloves and eye protection and have good ventilation when using this product.
Cleaning Build Up from CabinetsCleaning the cabinets will take some time but give this a try. If the cabinets are extremely soiled, and this isn’t working well enough, you may have to think about stripping, buffing, and refinishing.
Turpentine and Linseed Oil WashWear rubber gloves to keep your hands from getting irritated and for protection against the hot water.
To 1 gallon of hot water, mix 2 tablespoons gum turpentine and 4 tablespoons boiled linseed oil, stir together. (The turpentine will clean away oils, built up waxes or polishes and the linseed oil will replace the oils.)
Soak a soft, clean cloth in the solution then wring it out and wash the cabinets. Rinse the cloth in plain hot water and repeat this part of the process if necessary.
When you are finished wipe and dry with another soft, clean cloth and buff out. Apply surface oil or polish, if desired, when finished.
http://extension.usu.edu/files/publicat ... HI-500.pdfMrs. Clean