Use a digital multimeter, set it to AC volts, and connect it to the battery with the vehicle running. Any AC voltage above 0.09V means the alternator is bad.
This is called an AC ripple test (and IMHO your auto store should have done this, it’s not exactly hard).
Edit: other tests to do can include:
Testing for DC volts at the battery + and – terminals while running , you should have no less than 13.4. Less means it’s not charging, which can be caused by the AC ripple, or resistance (see below). More than 15.8 can be caused by a bad voltage regulator (often built into the alternator), which is bad for the electrical system in your car.
Test the voltage drop on both the positive and negative battery cables (which tests for high resistance in the electrical line being tested). The tl;dr is to test for DC volts with the car running – first negative battery terminal to some point on the outside of the alternator body, then positive terminal to the red wire on the alternator. You should not get more than a few tenths (0.01s) of a volt in either test; if you do, there is too much resistance somewhere in the system, probably from corroded or loose connections somewhere.
Test to check battery same tool- car off place voltmeter on DC note the voltage across battery. While leaving voltmeter leads on battery turn on the headlights, (car is still off) a huge sudden drop in voltage or fast draining is a good clue the battery is bad. A slow drop in voltage probably means the battery is ok. Headlights are a great load source to check the battery.