What is the black dirt looking substance that forms on the skin of my lemons?

Hi Claire, it sounds like your tree has sooty mould. This will be due to excretion from an insect possibly scale, aphids or mealy bug on the tree. Spray (with warm soapy water, or check at your local garden centre for a suitable spray) to control the insects and the sooty mould will go away. Any fruit that has the soot on it is ok to eat; simply remove it by washing the lemons, or just use it for the juice. To help grow successful lemons make sure you feed them in spring and summer with Tui Citrus Food and keep them well watered. Applying Tui Organic Seaweed Plant Tonic regularly will also help keep the tree healthy. Add a layer of mulch around the base of the plant too.