You need a digital voltage meter to test the charging system yourself. Here's what you need to do.
1. Engine off - battery standing voltage.
With the engine off and fully charged battery, hook up the meter directly to the battery terminals(not clamp, bolt, nut, etc...). Read the meter as volts DC. 12.8 volts is 'perfect' for a typical fully charged wet cell battery in new condition at room temperature. Be aware battery voltage will decrease with temperature.
2. Engine running - lo idle - no load charging voltage.
Engine running at idle, all lights, accessories and AC off. Read the meter after voltage stabilizes from starting engine. Usually 0.5 to 1.0 volts higher, typically mid 13 volts.
3. Engine running - hi idle - no load charging voltage.
Engine running at 1500 RPM, all lights, accessories and AC off. Read the meter. Should not be over 2.0 volts above battery standing voltage.
4. Engine running - full load charging voltage.
Engine running at 2000 RPM, lights on high beams, AC on MAX(blower on high). Read the meter. Full load charging voltage should be a minimum of 0.5 volts above standing voltage of battery.
Also note that a hotter alternator will decrease voltage output.
Another good test is to check the voltage with the negative lead on the negative battery terminal and the positive lead on different points starting from the alternator, fuse box distribution point and then the battery positive post.
If the battery is not sealed, inspect the cells making sure each is up to the minimum level. During charging, observe the bubbling in each cell. A bad cell will have little or no bubbles compared to the others. Observe all proper safety precautions at all times when testing and charging any lead-acid battery. You can also test the electrolyte in each cell with a tester available at most parts stores.
Never believe those 'charge indicators' on sealed batteries.
The only 'proper' battery test is a load test on a fully charged battery.
A load test on an alternator is best as well if available.