Bonnie: rent a power-washer!
Try a power washer.
I would mix up a bucket of soapy water....say a couple of quarts and about 1/2 cup of DAWN DISH WASHING LIQUID and give it a good scrubbing. Then rinse with hose. This method will not remove any of the color from your paint. You can use a plastic cup or glass to pour some water up on the door.
HINT: Go to the Dollar Store and buy yourself a broom with stiff bristles to scrub with. It sure helps on your back. I use one to clean my bathtubs...so I don't have to bend over! It works great. I keep this one for cleaning and do not sweep with it!
Sounds like mildew that forms when a surface doesn't get sunlight. Get some TSP from big box store, mix up a gallon and add 2 cups of bleach and a squirt of dish soap. Scrub it down with a nubby cloth or scrub brush. Rinse with a garden hose.
If you use a power washer make sure it is on low pressure or you could gouge the wood. Best to use a homeowner grade pressure washer for cleaning any parts of the house itself.
Ultimately, the safest way to clean the exterior of your old home (especially wood houses) is to use a regular garden hose and spray nozzle along with an extension pole with a nylon scrub brush.
Power washer works the best. I spray the door with a bathroom spray then blast it with a power washer. They are reasonable now and worth the investment.