Symptoms of bad battery or alternator?

Take it to a parts shop that has a roll-out or hand-held alternator/battery tester. They'll check the battery with the car off to see if it's at or close to 12.5 volts - if not, remove the terminals and check it again to verify that something isn't draining it while it's not running. If that's kosher, they'll have you start it and see if it's putting out in the ballpark of 13.5 to 14 volts while running - if it's only getting 12.5 or less while running, your alternator probably isn't giving the battery a charge.

Either way, I'd say it's safe to say that your battery is toast right now, because you can only run them down completely so many times and recharge them before they just up and totally die. With a fresh battery thrown on there (clean your terminals and put on some protectant, also a couple of felt washers is a good idea), you can get a more accurate test to see if your alternator is doing its job, too.