Carbon stains can be very stubborn to remove as carbon particulates bind with the fibers in laundry. Carbon black, found in India ink and some permanent markers, is likely to make a stain so ingrained that you will not be able to remove it, but milder carbon stains, caused by smoke and exhaust fumes (if you live near a road), for instance, can be got out of laundry with some careful treatment.
Using Dishwasher Detergent and Bleach
Check that the fabric is washable, not dry clean only.
Dissolve 1 cup of dishwasher detergent in 8 pints of water. Soak the fabric in the solution overnight.
Soak the fabric (if safe for bleaching) in chlorine bleach for 30 minutes before washing.
Wash the material according to the manufacturer's instructions.
Using a Solvent-Based Cleaner
Wet the whole area affected by the stain with a solvent-based emulsifier like those often used in dry cleaning. Pay particular attention to the most deeply stained areas. Spot treat those areas heavily with the solvent-based detergent.
Leave overnight, if possible, or for a couple of hours at least, to let the detergent interact with the stain.
Wash the fabric as specified in the manufacturer's instructions.
Repeat the process if the stain is not fully removed.