I also suspect, based upon the age and type of bike, that you have a bent derailleur hanger.
Here's the good news: 5-speeds aren't nearly as picky about perfect hanger aligment as 9 or 10-speed drivetrains. I'm pretty sure that you can do this yourself.
1. Shift your bike into a gear that makes the derailleur arm point straight down. Prop your bike up vertically against a wall or something.
2. Now look at your derailleur from the back. If the arm seems to be pointing toward the rear wheel - that's it.
3. If you find that you can't bend the derailleur hanger back with your hand, look for a 15" adjustable wrench. Clamp the wrench OVER the rear derailleur and bend it back until it's straight. IMPORTANT: Resist the urge to remove the derailleur so that you can bend the hanger by itself. If you happen to ovalize the derailleur mounting hole in the process you will be well and truely screwed.