Is there any chance you've cross-threaded it? If you're sure it's not cross-threaded, then I'd check for a missing washer or O-ring. It may have blown out of the aerator when you were backflushing it with the hose.
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
The type of valve that has to be "tightened down" to make a good seal is likely an older valve like the one in this diagram from ThisOldHouse.com:
Over time, there is a washer on the bottom of the stem that wears out. Forcing the handle tighter can fix the issue, but this ultimately just destroys what's left of the washer even faster. Normally removing the stem and replacing the flat or cone shaped washer is all you need to do.
Another problem that can occur is the metal "seat" for the stem that stays in the faucet when the stem is removed can become corroded to the point of not providing a good seal. The seat can be replaced (if you can find one), or they can be ground down with a special tool to provide a new surface for the washer to seal on. Feel the seat with your finger to see if it's smooth. Try to just replace the washer first, and only mess with the seat if the drip continues or returns after a short time.
They might be a challenge to find, but you can normally get new seats, an entire stem, or any washer you need to repair the faucet. A whole new stem will cost on average $15 or so, and they are easy to replace. You shouldn't need to replace the faucet unless the exterior is in equally bad shape and you just want something new.