Summarize:

Start by spraying the brick surface with a hose—water is typically effective at removing most dirt and dust from bricks. For tougher dirt and salt deposits, use a stiff-bristled brush dipped in a soap water mixture. If this isn't doing the trick, mix 1⁄2 cup (120 mL) of trisodium phosphate into 1 gallon (3.8 L) of water to create a cleaning solution and dip your brush into it before scrubbing the brick.  Apply a solution of 1 part bleach and 3 parts water to mildew and let it sit for 30 minutes. Afterward, scrub the surface with a wire brush. Consider renting a 1500 PSI pressure washer from a home improvement store for larger areas of brick. Never clean brick with acid cleaning solutions or you might ruin the paint job. Place unfolded pieces of newspaper over your doors and windows and tape them to the perimeter with painters tape. For larger areas, use multiple pieces of newspaper. Be sure to cover their entire surface and fasten the newspaper firmly with painter's tape. Protect any other areas that you do not want to paint—such as gas meters—with painter's tape or newspaper fastened with painter's tape. using acrylic caulk. Insert a scraper into the cracks and lever them to open them wider. After, brush out the dust with a straw cleaning brush. Cut your caulking tube at a 45-degree angle, taking care to keep the hole size small – no further down than 1⁄4 inch (0.64 cm). Insert the tube into your caulking gun, hold the release trigger, and pull the steel rod back as far as you can. Now, press the trigger and move the tip steadily through the crack until it's filled.  Allow the caulk to dry for about 5 hours. Press the caulk flat into the cracks with a putty knife to even it with the wall. Be sure to move the gun in steady motions. If you move too fast, the caulk will be thin and uneven, but if you move too slow it will be clumpy and difficult to work with. Dip a paint roller into the primer 2 to 3 times. Start around 1 foot (0.30 m) from the bottom of the wall and about 6 inches (15 cm) from the corner. Aim your roll upwards at a small angle while applying light pressure. Once you reach 2 to 3 inches (5.1 to 7.6 cm) from the top of the wall, roll up and down while moving back to the corner. Continue moving from the floor to the top and always move ¾ the width of the roller so each stroke overlaps.  Continue rolling until all of the bricks are covered. Put a few extra coats of primer on the areas that were affected by efflorescence. Wait for the primer to dry—the recommended drying time should be listed on the package—before moving on to the next step.
Clean the bricks thoroughly with a cleaning mixture. Cover the windows and doors with newspaper. Repair wall cracks Apply latex primer to the surface of the bricks.