From the way you describe it, it sounds like it's a regular gate valve. You might be able to repair it as you describe by removing the packing nut and replacing whatever is broken or stripped out inside.
The only exception to that would be a ball type valve, but those will just have a handle that makes only a 1/4 turn. Those generally aren't repairable, but then again they almost never fail either.
Personally, I would just remove and replace the gate valve with a good ball valve. Gate valves have a tendency to rot out or seize up from lack of use. I actually had one once that disintegrated in my hand when I tried to turn it off (spraying water everywhere and making a nice mess). Ball valves are just much more reliable.
I assume that this is leakage that happens with normal use. I have found that with many single handle faucets, if you push the handle all of the way open, hard over and hold it there, they will all leak.
Turn off the water to your faucet from under the sink. There should be a shutoff down there. Make sure that both the hot and cold water lines are shut off, so that when the faucet is open, no water should come out.
Open up the faucet by removing the handle. Usually this is easily done with a screwdriver. There will be a screw someplace to do this with, although it might be an allen type set screw. Sometimes that screw is neatly hidden underneath a plastic plug.
Pull out the cartridge. Drive to your nearest home center, with the cartridge in hand. Now it will be trivial to find the exact cartridge. If you are still unsure, ask for help.
Buy the proper replacement cartridge, complete with all o-rings, etc.
Replace the cartridge into your faucet with the new one, replacing ALL o-rings, all of the pieces that come with it with the new ones.
Put the handle back on, turn on the water.
If it still leaks, buy a better grade of faucet. I have found that cheap hardware sometimes really does perform like CHEAP hardware.
Best Answer
Woodford Faucet is probably the place to go for the parts breakdown if I can read the handle right. They produce a kit specifically for repair. You'll have to look for better identification, but here's the Model 17 kit (round handle) which probably addresses all the issues.
American MFG that's been around like forever and has always been good about repair kits. You'll recognize their key'd faucets if you've used campsites.
EPDM Packing that's under the Nut