Although 3/4 cup of bleach followed by a thorough flushing of hot water can be used to deodorize drains, it won't help a clog. Bleach is an excellent sanitizer but won't eat through the hair, washcloth bits, toothpaste and other matter that gets trapped inside pipes and causes clogs. Many people find this confusing since bleach is an ingredient in many drain-cleaning products, but the bleach is combined with other cleaning agents and caustic materials in these products -- they do not rely on bleach alone to unclog the drain.
A single ounce of straight bleach requires dissolution in 312,000 ounces of water to be safe for fish, so protect the environment by limiting your bleach use when possible. There are several ways you can attempt to unclog a drain on your own before calling a plumber for help, though bleach should not be one of them.
Water, Baking Soda and Vinegar
When a drain is clogged, the Washington Toxics Coalition recommends pouring a tea kettle full of boiling hot water down the drain. You can also try placing 1/2 cup of baking soda in the drain followed by 1 cup of white vinegar and 1 cup of warm water. This creates a bubbling action that will help clean the entire pipe. This method works well for clogs caused by toothpaste and soap residues but may not be enough to dislodge hair and other tougher clogs.
Take the Plunge
The right plunger can move a clog in a sink just as it can in the toilet. To try plunging a clogged sink, first remove the stopper from the drain and block any overflow openings with a damp rag or paper towel. Spread a little bit of petroleum jelly onto the lip of the plunger to create a better seal, run a small amount of water down the drain and then plunge the sink three or four times. Run a bit more water down the drain and plunge again, repeating as needed until the clog is removed.
Call a Plumber
You can try and if you have experience with plumbing, but both of these methods require removal of the P-trap and proper resealing and re-installation to prevent leaks. If you can't remove the clog on your own with the plunger, it may be best to leave the job to the professionals rather than create a leak or risk causing other problems.