Every once in a while, if a sink clog is bad enough, you will have to remove a clog from the sink trap itself. The sink trap is the 90 degree bend in the sink piping directly below the drain.
Step 1 - Remove the Trap
Before removing the sink trap place a bucket or large plastic bowl directly underneath the sink trap in order to catch any flow of water remaining in the sink or trap. Once the container is in place, use your pipe wrench to unscrew the trap coupling at the sink drain and also at water pipe going into your wall.
Step 2 - Remove the Clog
You can use a wire hanger or even a drain auger for this step. Push the wire hanger or auger into an open end of the removed sink trap and work loose the clog. Once the clog is clear you may want to send the auger into the pipe attached to the wall just to make sure that there are no further clogs down the line.
Step 3 - Reinstall the Trap
Using your pipe wrench, reattach your sink trap at the wall and sink drain. Make sure the coupling is snug but do not over tighten as this will risk cracking your plastic drain pipe.