So start it up, then while its running disconnect the positive side of the battery, if it keeps running then the battery is suspect, if it dies then the alternator is not charging.
NO NO NO NO NO NO NO NO NO NO NO NO NO
Dont EVER EVER do that. The alternator will go into full field and you'll fry your rectifier diodes if you pull the +ve terminal off the battery while its running. In a genny you may damage the voltage regulator!!!!!!!!!!!!!!!!!!!!!!!
The charging system is actually a battery \"MAINTAINER\" it was never designed to fully charge a low battery. Its suppose to top up the battery with voltage and current (alt/genny assumes you have a perfect battery)when it reaches a threshold (all designed within the regulator) it will charge the system.
Having a POS battery (which most people do) will make the charging system cycle alot more than it should so this is why it causes more wear and tear on the alt/genny.
If your a chevy owner a crappy batter will screw up that junk delta wound alternator (maybe \"y\" wound) real quick. This is why many of them change the alternator at least 4-6 times in the liftetime of that car (chevy) If you own a honda you may change the alternator after the car hits 200,000+ Km's.
A VAT40 or a minivat from snapon can quickly test your system.