Why does water and gunk back up through the overflow hole in the bathroom sink?
I would start by removing the trap, putting a bucket underneath the tailpiece (below the sink) and just running some water and seeing what happens.  Clean out the trap while you're at it and make sure there isn't a mess of hair stuck in there.  If it all comes through without a problem and doesn't back up, then I would suggest buying a snake and going through the wall pipe and seeing what you can pull out.  I've always been able to get a slow drain going with a snake.  It might take a few tries, but it can usually work out.  Make sure that you push the snake right up against the pipe opening, and then just keep cranking until something changes.

You want to be careful with chemicals because they can discolor the chrome around the sink if you're not very careful, and they can also react with the pipes in a unsafe manner.  They make some gels that are more organic in nature that can eat away at organic clogs (hair, etc) that are much safer to use.