How do I get rid of this clicking sound in the back wheel? Difficult to diagnose, I know.

Basically, I bought a single speed Schwinn a few months ago, and after riding it for a couple of weeks it started clicking in the back wheel somewhere.

I've taken it in for a tune-up, mentioned to them there was a clicking sound, but they didn't notice anything (or didn't pay attention), and it still clicks.

It doesn't click, however, if you just pedal it with your hands, but seems to have something to do with putting your weight on it when you're actually riding.

Should I just take it in again? Or is there something cheaper I could look at?
The problem could be the lubrication on the rear hub. Perhaps it wasn't well greased at the factory. A tune-up should have caught this problem, particularly as you mentioned it. Another possibility is that there is a ball bearing missing from the rear hub, so that once every time the wheel turns around, the axle falls into the small gap created by the missing bearing (this happened on my daughter's new bike after about 3 months of occasional riding).

If the bike is still under warranty, then I would take it in. The problem should have been fixed the first time.

If you want to try fixing it yourself, you will have to take off the rear wheel and disassemble the hub. This isn't hard, but if you've never done it before, try finding a local bike shop that will show you how. Some towns have co-op bike repair shops where for a small fee you can use their tools and there will be a mechanic on hand to guide you.
(for example Ottawa - http://www.re-cycles.ca, Waterloo - http://www.theworkingcentre.org/ct/cycles/cycles.html) Try looking at this wiki for a shop near you.