First job is to remove the debris. I have found that using a pair of large wire cutters works really well for screws. You can slip the under the head of the screw when the screw is pressed flush into the tread. Have that reamer handy though because the moment the screw comes out, the air inside your tire will follow it.
Before you plug the hole, you will need to ream it first. This does two things. First, it makes the hole large enough for the plug and plug gun to fit through. Second, it gives the plug a rough surface to grab on to, increasing the likelihood your plug will take hold. The technique is simple. First, insert the reamer all the way into the tire. Then work it straight up and down a few times, until you feel the reamer move a bit more easily. This enlarges the hole. Once you've reamed the hole, spin the reamer a few times in the direction of the threads of the reamer. This roughs up the inside of the hole so the plug has more surface area to stick to.
Next up is the plug. First, load the plug gun. Pull the plug through until there is an equal length on either side. Tire Slime's plugs are really long. You could actually cut them in half and they would still work just fine.
Now it's time to plug that damn leak! Stick the plug gun in and push. The biggest trick is not to push the plug in too far. This is the MOST IMPORTANT STEP. Leave about 1/2″ to 3/4″ hanging out of the tire. It will eventually wear down once the plug vulcanizes.
Since this was an especially long plug, I left a little extra to make it easier to see and then trimmed it to size after I snapped this pic. If not enough length is left on the plug, it will suck into the tire.