My city bike, which is a big hybrid, also sleeps outside, and is quite rusty. I oil it once in a while - which makes it look better - but I haven't bothered cleaning the chain.

DISCLAIMER: I'm no bike mechanics.
If you really wanted to clean the chain on the cheap, you could use WD40 and a good brush with hard bristles (found in any bike shop). A tooth brush is not bad but I usually find them too soft. (Maybe if you buy the "hard" style it would work better?). Make sure you rinse with water when you're done and oil very well.

I wouldn't use WD40 on expensive road bike components, though.