You can do a variety of different things to unblock a kitchen sink according to what the stoppage might be. I have been know to grab a plastic glass with an opening that will fit over the drain and causing a suction move the cup up and down until the sink unclogs. This method does not work all the time and can be quite gross on your hands. All the junk trapped in the drain can and will come up into the sink many times before draining. 

The proper way to do this type of method is with a plunger. All you do is position the cup of the plunger over the hole of the drain. Now firmly move the handle up and down a few times without moving the cup totally away from the drain. It should unclog the kitchen sink in no time unless you have a huge clog, which will take a more drastic method. 

You may have to unblock the trap, which is a bit more troublesome. All sinks have some sort of trap that blocks odors. The most common is the tubular trap. Before you begin, you should place a bucket under the pipe so the excess water will not go everywhere under your sink.

First, look for an access point. If you have one on your tubular trap, you are in business. All you have to do is remove the cap. According to how tight the cap is on you may be able to remove it with your hand, if not you will need a wrench. After you have loosened the cap with the wrench, use your hand again to ensure you do not strip the threads. Once the cap is off all you need is a wire such as a coat hanger. Place it into the tube and pull out any debris. Once done, put back in place. 

If you do not have an access point for your kitchen sink, then you are in for a bigger job. You will need to remove the nuts and bolts above the U-bend. When you have removed the nuts and bolts, gently remove the U-bend and clean it out with a wire. Be sure you have that bucket underneath. Once you have cleaned it, replace it. 

Now you can unblock your kitchen sink and use the money you saved from paying the plumber and go out to dinner.