The problem is that parts-store alternators aren't nearly as durable as the factory one was, so unless it totally dies, I'd keep it if I were you.

Hook the multimeter up to the battery so you can see the display from the driver's seat. If it drops noticeably when you try to crank it, the battery likely needs replaced. After that, if there's no click, the likely suspect is either a loose or corroded battery cable going down to the starter, the tiny wire with the plastic clip on the side of the starter solenoid is loose, or the starter motor is just plain bad.

Check for the first two: Physically go in there with a wrench, loosen, clean & inspect, and tighten the battery and starter ends of the big wire with the rubber boot, and the starter end of the small one with the plastic clip.

After that if there's still no cranking with a good battery, it's likely a bad stater motor. <img src="/forums/images/graemlins/wink.gif" alt="" />