Simply unscrew the aerator from the tip of the spout. Most are attached by hand, so they can be easily removed. Ensure you dry your hands and the faucet for a better grip. Try pliers or apply heat (hot water, hairdryer, etc.) to loosen it if you cannot remove it.
Once you remove the aerator, gently separate the parts. Remember how everything goes back together. Use a small brush to remove any built-up dirt and debris. A good soak in some vinegar and baking soda should remove any stubborn grime.
You may need to replace the aerator at this point. If the screen or any other parts look damaged or rusty, you can find a variety of replacements at your local hardware store or call a plumber to fix it.
Once clean, reassemble, and reattach!
Remove the tap handle with a pair of pliers
Coat the handle stem with some plumbers grease
Reassemble.