Sooty Mold on Lemon Trees
It sounds like your lemon tree is suffering from sooty mold. This is a fungus that appears as black speckles on the tops of leaves.
Sooty mold thrives in the secretions, or honeydew, of common pests such as mealy bugs, aphids, whiteflies and scale. Honeydew is a sticky substance, which attracts windblown sooty mold spores.
The first step you should take is to check your other houseplants for insect infestations and sooty mold. Then you should isolate your lemon tree and any other contaminated plants to prevent the problem from spreading.
Next, with a soft cloth, gently wash the leaves with a 70 percent isopropyl alcohol to remove the mold and honeydew. Complete the process by rinsing with water.
Now you should tackle the insects, which are really the source of the problem. Because you mentioned "little white bugs that fly," I'm going to assume that you have whiteflies.
You can confirm that you have white flies by doing a simple test. Just shake the foliage and if a cloud of white insects fly out then you've spotted the culprit. The only way to keep the sooty mold from recurring is to eliminate the whiteflies.
To control whiteflies I suggest repeated applications of an organic fruit and vegetable pest spray. It should be organic because you want to be able to eat the lemons later.
Now, for this to work best, you really need to soak the under side of the leaves and the top of the plant all the way down to the base.
To help prevent the return of these pests hang a whitefly trap on a branch of your lemon tree. This is an adhesive strip that is yellow in color. The bright color is attractive to whiteflies and other insects so they're drawn onto it, stick, and die.
You can pick these up at your local garden center and they're easy to apply.