I have an exterior faucet that is leaking badly. Water is coming from the faucet handle, not further back. I have tried to remove the screw from the faucet face, but wonder if I should just use a wrench and remove the whole thing from the pipe? Do I need to cut off the water supply first? Looks simple, maybe not?

If it is a very old faucet, replace it--yes, cut-off the water supply to the faucet.

XenoSapien

You really should be able to unscrew the nut and replace the packing underneath. Once you have it apart, you should be able to go to a good hardware store and buy the correct packing. Does it just screw onto a pipe sticking through the wall, or is it a frost proof one extending inside? Either way, with the water shut off, you should be able to remove it and replace it. It might be easier to get a new frost proof one started if you have access to the joint inside.