What could cause this rhythmic clicking/drumming sound in my bathroom sink pipe?
Most bathroom sinks have a pop-up stopper, which works well for stopping the drain when you want. Unfortunately, it's also great at collecting hair and other debris that goes down the drain.

These stoppers should be removed and cleaned from time to time, to make sure the drain is flowing properly.
Clean the Stopper
Cleaning the stopper is the first step in troubleshooting a slow drain. There are many types of sinks, drains, and stoppers.  I will cover cleaning one of the more common stoppers.
Removing the stopper
If you look under the sink, you'll notice a pipe coming out the bottom. On the back side of this pipe, you'll notice a rod sticking out.  Where the rod enters the pipe, there should be a nut.  If you remove the nut, you'll be able to slide the rod out.  Once the rod is removed, you'll be able to pull the stopper up and out of the drain.
Clean the gunk
Once the stopper is removed, clean and rinse it.  DO NOT rinse the stopper in the sink you removed it from.
Install the stopper.
Place the clean stopper back in the drain, and align it so that the rod will fit in to the hole at the bottom of the stopper. Go back under the sink, and reinsert the rod. Actuate the rod, to ensure the rod is properly inserted into the stopper (the rod should move the stopper up and down). Tighten the nut back down. Actuate the stopper a few times, to make sure the nut is not too tight or too loose.  If it's too tight the stopper will not move, or will be difficult to move. If it's too lose the stopper may not stay in the full raised position.
Clean the Trap
The next step in troubleshooting a slow drain, is to clean the trap. Again drains may vary, I'll only be covering a trap with hand-tightened compression fittings.
Remove the trap
Since there will be water in the trap, you'll want to have a bucket under the trap before disassembling it. While holding the trap in place, remove the compression fittings at both ends.

Once the trap is free, carefully pull it free from the other pipes and dump it in the bucket.
Clean the trap
Clean the trap using paper towels and hot water.
Install the trap
Once the trap is clean, installation is the opposite of removal.

If after you've cleaned the stopper and trap, you don't notice an improvement in the drainage, or you're still having trouble. It might be time to contact a plumber, and have the line snaked out.