Is my battery dead, or do I need a new alternator?

most definite the charging system. if possible check the output voltage for 12 volt systems it should be charging 14.5 volts . check at the alternator post, then check at the battery connection.. if voltage is good at the alternator and not at the battery look for loose connections. Clean terminals on terminals, also check drive belts make sure they are not slipping no/low voltage at the alternator the alternator is faulty.