Deep scratches in a black car are extremely noticeable and tough to repair, but not impossible. The thing that makes deep car scratches difficult is the bodyworks garage may charge you much more to fix the scratch than you feel the job is worth. If you feel comfortable, you can do the job for next to nothing, but care and precision are of utmost importance during the process.
Mix one part water and one part white vinegar in a plastic bucket. Apply to the scratched area with a sponge and scrub firmly but not over-aggressively. Use a toothbrush to get between the cracks. Rinse with clean water.
Dip the 2,000-grit sandpaper in cold water and soak for a few minutes. Remove the sandpaper and lightly rub in a circular motion in the cracked area. Be careful not to remove any of the non-damaged paint.
Put a small amount of paint on the end of the toothpick and carefully spread in a thin, straight line along the crack. Stay in the lines and don't put too much on in the first coat.
Apply layers once per day or once in the morning and once in the evening until the crack/touch-up paint has become even with the rest of the paint. Once the last layer of touch-up paint is dry, begin applying coats of clear touch-up using the brush that came with the paint. Paint the clear touch-up over the crack and a small area around it to blend the old and new paint jobs. Do this until the clear touch-up is a little above the other paint.
Lightly sand the touched-up area with 2,000-grit sandpaper until it is even with the rest of the paint. Rub on the car polish with a firm sponge in small circles in all directions to bring back the shine. Do this slowly and gradually, over several days.
This series of steps will not make the area completely invisible, but it will mask it considerably.
Do not "paint" the touch-up paint over the scratch. The process is actually gradual and delicate. It requires you to fill the crack slowly and thinly. Begin painting when you get to the clear touch-up step. This is how you blend the new paint and the old paint.