The salmon should be laying on a cutting board with the backbone parallel to you. The head of the fish should be facing the hand holding the knife. Cut behind the head and gill plates on a downward angle away from the body using an 8-inch flexible filet knife. Without removing the knife from the fish head, turn it to face the tail, keeping the handle lower than the tip of the blade.
Slowly and carefully run the blade against the backbone through the length of the salmon, while trying to not saw through the fish. Once you have freed it from the bones, lay it skin side down on the cutting board. Use both hands to do this.
Without turning the filet over, insert the knife just under the bone structure. Use the same slicing motion with your hand flat on the fish for stability and run the blade again through the length of the fish.
To remove the belly bones, make small smooth strokes underneath them and they will cut cleanly away. Next, run the blade underneath the remnants of the backbone line, lifting as you cut. Now hold the tail firmly in your free hand, insert the knife between the skin and the flesh, and pull the skin taut as you cut the salmon meat free from the skin.
Lastly, if you run your fingertips lightly over the filet, you will feel the pin bones protruding up. Using your bone tweezers, pull the bones out in the direction of the head. This will cause less tearing of the fish. Use the bones and small leftover pieces to make fish stock.