Start with a sharp knife. It is difficult to skin salmon with a dull blade. Hold the edge of the filet with your fingers and slide your knife between the skin and meat at about a 45° angle facing the edge of the blade toward the skin. Position the blade so that the fish is in the middle. Do not move your knife back and forth to separate the skin from the meat. Instead move the salmon back and forth on your knife blade holding the skin. Keep the edge of your blade at an angle so it cuts between the meat and skin without cutting through the skin. This takes a little practice, but with a sharp knife it is actually quite easy.
To remove bones, run your fingers over the top of the fish. You will find a line of bones. Remove them one at a time with a pair of tweezers, pliers, or your fingers. Pull them out going with the grain of the fish so they slide out without tearing the meat.