Step 2: Pre-cleaning preparations
With a snapper, you can usually get them to bite onto that stick you should have, then they'll hold on pretty hardcore. Pull the neck out as far as possible, and then, cut the head off, leaving as much neck as possible. If it won't snap onto the stick, then just wait for it to poke it's head out enough.
The key is to cut off the head cleanly, in one chop. A hand-axe would help greatly, especially if you positioned the turtle's neck over a board. Minimize the animal's suffering. One thing my dad always taught me when we were fishing, was that you never mock your catch, and if you're going to kill it, you should do it as painlessly as possible for the animal. Do turtles feel pain? I don't know. But it'll be better for you (less gore, less mess) and the turtle (suddenly over, no bleeding in the face or whatever).
Now, take the pliers, and hold the wire with them. Poke the pointed end that you clipped into the wire through the turtle's tail, or (if a soft-shell), it's shell at the very tail-end. Wrap the other end around a tree branch, and leave it for a while, so it will bleed out. Small turtles= less bleeding, I would assume. So, no set time. I bled mine until the blood stopped dripping and started clotting on the neck.