Grab a cheap digital voltmeter and measure the voltage across the two batery terminals while the engine is running. It must be between 13.75 and 14.75 volts. If it is around 12.0 to 12.4 volts, the charging system is indeed dead.
If it's dead, you'll need to measure the voltages on the two small wires plugged into the back of the alternator. One must have full battery voltage. The second one will have less, but it must not be 0 volts. Those voltages will only be there steady when the engine is running. Those two readings will determine the direction to follow next. The possibilities on the second wire are 0 volts, exactly the same voltage on both wires, or something in between.