Apart from warning lights on the dashboard, the easiest way to diagnose an alternator problem is dim lights. If the lights get dim while the car is running, there is probably an issue with the alternator.
Alternators play a big role in a car's electrical system. Part of this role is to provide power to the electrical accessories in the car, including the lights. So if the lights are dimmer than usual, or if they start bright but quickly start to dim, the problem is likely to be with the alternator rather than the battery.
The are other indicators that can be used. If a fully charged battery is installed in the car, but the car still struggles to start, there might be a problem with the alternator. Also, many cars have warning lights that appear on the dashboard when problems start developing with the alternator.
One final method of checking whether the problem lies with a battery or an alternator is to use a voltmeter. This is an inexpensive piece of equipment that can be bought at most car accessories stores. With the engine running, the voltmeter should show an output of between 13.8 and 15.3 volts when connected to the battery. If this is not that case, the alternator may be faulty.