Most of the citrus trees at my grandmother's house are sick like this, and have some kind of black mildew on their leaves and fruit, and the fruit is frequently scarred. They are getting old, but I think they might have some kind of disease.

The damage looks like it is from Citrus Rust Mites. Wettable sulfur should help control them in the future. I don't know the rate, so make sure you read the label.

The black mold covering the leaves and the stems is called Sooty Mold, it will rub off with your fingers or finger nails.. It is a mold that grows on the sugary excretions from insects . A thorough spray with a good horticultural oil such as Ultra Pure, Volk Oil, or even Neem Oil will help to get rid of it and also many of the affending insects at the same time. - Millet

I never saw them with my own eyes, but couln't it be the California Red Scale, Aonidiella aurantii?