This damage is most likely caused by an insect called citrus whitefly. The insects feed on the leaves of the plant and as they do, sap drips down on leaves below the feeding area. When that sap (technically referred to as honeydew) sits on the leaf surface for an extended time, those leaves develop a black powdery substance called sooty mold. The insects can usually be controlled by spraying with an insecticide that targets soft bodied insects, such as insecticidal soap or horticultural oil. It will also help to cut back on nitrogen fertilizer, if you are using any, and perhaps to do some selective pruning from time to time, so that grown is not overly dense.

http://ipm.ncsu.edu/AG189/html/Citrus_Whitefly.HTML