Remove the tube. (While it is technically possible to repair a tube without removing the wheel, it is much easier to work with the wheel off the bike.) Make sure to inspect your tube for any debris.
You should be able to feel and/or hear most small leaks. For really hard-to-locate holes, immerse the tube in water a section at a time and look for a tiny stream of bubbles. Some flats are caused by pinching the tube against the rim. This usually creates a pair of punctures, or a “snakebite”—make sure you find and patch both.
Prep the surface. Most patch kits come with sandpaper, use this to rough up an area around the hole a few millimeters larger than the patch you are using. If you have a choice, select the smallest patch that covers the hole. If sandpaper isn’t available, a rock or any other rough surface can be used, just make sure there isn’t any debris left on the tube.
Apply the glue. Apply a thin layer of vulcanizing fluid to the tube, enough to cover a spot a few millimeters larger than the patch.
Wait five minutes. Find something to do. Get out your smart phone, look at funny memes, call your mom, roll a cigarette, take a quick nap, but DO NOT skip this step. Getting that molecular-level bond depends on giving the glue enough time to set up.
Apply patch. Peel off the foil, center the patch, and press down firmly. Leaving the clear plastic on the non-business side of the patch keeps it from sticking to the inside of the tire, and keeps the glue off your fingers. With the tube on a flat surface, use something not-sharp (tire lever, screwdriver handle, rock) to press the edges of the patch into the tube.
Install and inflate tube. This is a permanent fix, and assuming you followed all the steps properly, there is no need to replace the repaired tube with a new one.
We hope you don’t have to do this often, but please use this as a reference; practice taking your tires on and off at home. This way you can sharpen your skills and be ready for when it happens on the road.
Graphics & Reference: http://bicycletimesmag.com/how-to-fix-a-flat-tire/