Cut off the tough green leaves. Don't overthink this! Cut off the dark green, and leave the light. (Try to do this on a surface that will be different from what you will use when you cut them after they are cleaned; otherwise, you'll get dirt on your cutting board.)
Stand up your leek, balancing it on the roots, and cut a horizontal cut all the way down without cutting through the roots. Repeat this with the knife verticle, this time forming an X down the center of the leak. (See number 3.)
Wash the leek under running water to remove all dirt.
Place your knife perpendicular to the onion, cut slices off it, and all the way down to the root. Discard the root. Now, use your leeks!