Turn the water off by closing the home's main water lines.
Pop off the cap on the faucet handle, and unscrew the tiny screw under the cap with your Allen wrench. Slide the handle backward and off of the faucet.
Unscrew the bonnet or dome cap under the faucet, if your model has one, by turning it counterclockwise by hand. Take the pin out of the top of the faucet with a pair of pliers. Simply pull it straight up and out of the faucet.
Grip the metal rod on the very end of the core with your pliers, and turn it back and forth gently to work the core out of the faucet body. If the core is broken, be careful not to pull too hard, as only part of the core may come out. Continue to work the core free until it slides out of place.