this...
A bad battery can kill an alternator by making it work more to try to recharge it...
A bad alternator can kill a battery by a)undercharging, leaving the battery to make up the difference while the car is running and thereby discharging it, or b) overcharging and cooking away the acid, causing damage to the plates...
Take the car to your local auto parts store... most will run a free charging system diagnostic for you if you ask. the system will be checked with the car off, and with it running at idle, and they will (should) ask you bring the rpms up to check it at "driving" speed.. also with all accessories off, and again with accessories on (lights, radio, heat, etc).