A clean brick makes a bigger difference than you'd think. Here's the most effective method, plus the mistakes most people make.
Thoroughly wet the brick with a garden hose. This prevents the cleaning solution from being absorbed too deeply into the porous brick.
Mix Turbo Clean Degreaser with warm water. Apply to the brick with a stiff brush, working from top to bottom.
For heavy stains, make a paste of baking soda and water. Apply to the stain, let sit 10 minutes, then scrub vigorously with the brush.
Rinse from top to bottom with a garden hose. Ensure all cleaning solution is removed.