First things first, you need to clear the debris that is blocking the drain. Use a plunger to clean it for the water to flow freely.
Boil 1 litre of water, add a spoonful of liquid detergent to it and stir. Pour the boiling water into the sink to dissolve the sludge and fat on the walls of the drain pipe. Repeat the process for a couple of more times to allow the drain pipe to run more freely.
Take a cupful of baking soda and pour it into the drain followed by a cupful of vinegar. Let the mixture sit for 15 minutes. Then pour a bucket of boiled water for the final blow.