Check the tightness of the fanbelt.
Check the output of the alternator at various rpm (there should be numerous threads describing the procedure, but I just tried a search, and the search engine leaves A LOT to be desired - didn't find one of the ones I thought I should). Basically, measure volts across the +ve and -ve terminals of the battery. At idle, should be around 14V. As you increase revs, should remain 14V. If volts increase, then the regulator is bung. And the resultant overcharging will eventually kill the battery.
If the regulator is ok, try turning on the headlights - a healthy alternator should be able to maintain 14V at idle.
Also, you should consider the age of the battery - they don't last forever.