What is the best way to clean this stone wall?
The best thing to use would be a pressure washer with a concrete cleaning solution in it. Those stains are algae or mildew stains. If you don’t have access to a pressure washer, something like Tilex might take the stains up (spray it on, leave it set for about 15 minutes, then rinse it off with a hose). One potential problem with that, though, is that the Tilex may wash into a planter and harm some plants. But, you can get it in half gallon bottles at the Home Depot. You might want to get a sprayer for it, though (about $15). Otherwise, your hands will get *really* tired spraying the Tilex over that large an area with a trigger spray bottle. I would suggest getting a small bottle first and trying it in a small, inconspicuous area first to make sure it works. If so, then go to town. You may find that you also need a scrub brush to get the stains up. You can get one attached to about a 6-foot pole at the Home Depot for a few bucks.