Black water/gunk that is backing up into your sink is a sign of severe blockage in your pipes. The darkness in the water is bacteria and other material accumulating in your pipes, which is why it's important to wear gloves and other protective wear when cleaning the sink. But before you go about cleaning, you need to get to the root of the problem by removing any debris from your system.
The process of removing blockage from your pipes is easy, but it could stem from a number of causes. The most common cause could be your drainpipe. This happens when layered dirt, soap, toothpaste, hair, etc. begin to build in the internal walls of the pipe, resulting in water backups. It is something that doesn't happen immediately, but over a period of months or years. The best way to remedy the problem is to use drain cleaner, but if you don't want to risk damage to your pipes, you can allow vinegar and baking soda to sit overnight. If this doesn't help, you may need to go in with an auger, or dismantle the pipes to search for the underlying issue, but a larger problem could be at hand.
Let's say you're on a sewer system, and you tried to unclog your pipes with drain cleaner, but to no avail. Shift your attention from the drainpipe to the branch pipe. The branch pipe is the larger pipe that connects to the sewer drainpipe. The sewer drainpipe is responsible for sifting wastewater from your home into the city sewer system. Check your toilet or bathtub to see if your fixtures are having the same backflow issues. You can remove the clean-out covers to see what's going on inside the pipe to see if the sewer pipe is flowing properly, but you can also start by removing the stopper and filling the sink with at least an inch of water. Use a plunger to try and push any blockage from out of the system. The water will be a good way to wash out any garbage that may have accumulated in the deeper pipes. If that wasn't enough, use a bucket to catch any water as you remove the pipe. From here, you're going to use an auger into the opened pipe to remove any deeper debris. This is an all-encompassing approach that will handle any drain, vent or branch pipe issues.
The vent pipe is responsible for ventilating the branch pipe. It also brings in fresh air into the plumbing system. If you chose to remove the pipes and use an auger, you can check to see your method worked by finding the vent pipe (usually on the roof) to see if the clog was removed. Usually the vent pipe sits directly above the bathroom. Use a flashlight and look down the vent pipe for any blockage. If you find anything in the vent pipe, you can use a tool to remove the debris, or use a hose to flush anything out of the system. Ensure that the hose you are using is high-pressure based to completely clean out the pipe.
If the problem persists, check with your local city to see if it is an issue with the public system. Sewer is considered more stable than septic, but there have been horror stories of sewage backing up into people's bathrooms from sewer lines. If you're dealing with a backup from a septic system, do the same aforementioned steps above, or the system may need to be emptied by a professional.