when the alternator fails, the red battery light on the dash comes first. You should be able to start the car on a fully charged battery with a failed alternator. So charge the battery and try to start it. watch for the red light on the dash. Then test the belt and the belt tensioner. Then if it is indeed the alternator, you may take it out and test at an alternator shop. It is usually voltage regulator that fails. $50 for voltage regulator and 3 hours of work to get the alternator out and put it back. Dealership will try to charge you about $1000 for installing a brand new alternator.