Match the color of the paint. Although it may seem like black is black, there are many variations in shading and gloss. You may need to purchase several different types of black paint until you find the one with the right finish. Test out your paints on an inconspicuous spot on the chair and choose the one that blends the best.
Use a fine or medium grade sandpaper to gently smooth out the surface of the chair around the chip or scratch. Do not extend the sanding too far -- you want to keep the job small. Remove any dust with a damp cloth and wait for the area to dry.
Paint the sanded area. Use a very small brush and paint in one direction. Fill in the entire area that you sanded and allow it to dry. You may need to repeat the process until the area matches the rest of the black paint.
Apply a thin coating of clear coat on top of the paint to help it last longer and bring an added gloss to the spot. Typically, most black furniture has a clear coat. If the surface of the chair is dull, however, you can skip this step.
If the area that needs to be repaired is large, or if you cannot find an exact match for the paint, you will need to strip the chair. Use a heat gun or a chemical paint remover. Follow the instructions and remove all of the paint from the chair. Once this is done, you can repaint it in the black paint of your choice.