The black discoloration is algae that has grown in the wet season and dries black. It is easiliy removed with a pressure washer. I recommend you consider purchasing an electric power washer at $130 or a more powerful gasoline powered washer at about $300. You can rent a commercial powerwasher for $65/day. The cleaning can be done is less than 1-hour with a pressure washer using no chemicals and requiring no scrubbing. It will deep-clean the concrete and restore the original texture and color.

Power washer is a very useful tool for cleaning around the house. Other than that you can try clorex bleach(for laundy) to remove the black algae stuff off the gutter pipes and roof over hung. I poured it in a paint tray and applied on the surface with a 3" brush. Then wipe off with a old damp cloth. Doing every 2' at a time.

Hope this helps.