Take a volt meter, when the car is running the voltage across the battery should be around 13.00-14.5V or somewhat in this range, when the engine is off, the voltage should drop to around 12-12.5V.

If you see the voltage change between engine off and on you can rule out the alternator as it is working fine and you should look elsewhere.

If it doesn't raise the voltage, it means there is no charge.