It's very easy and nothing needs to be disconnected if you are sure the battery is okay. With the engine/ignition off, measure the battery voltage. A good battery should be about 12.5 volts at room temperature. If the voltage is near 12 the battery is significantly discharged. If voltage is below 12 the battery probably has a shorted cell.
Now start the engine and read battery voltage at idle with accessories off. It should read 13.5-14.5 volts. If the voltage reading is 12.5 or less the alternator may not be functioning. If the voltage reading is below 13.5 bring up the revs to see if it increases. If so the alternator output is weak or the battery has a shorted cell. Check for a shorted cell by disconnecting the negative battery cable. Let the revs down, and if the engine keeps running, the alternator is okay.
The regulator should keep system voltage in the 13.5 to 14.5 range under all conditions except possibly a very heavy current load at idle, such as what may occur if virtually every accessory is on or a severely discharged or shorted battery, so this simple comparison test of system voltage with the engine off and on is how to test an alternator on the car.
The higher than battery voltage with the engine on places a "back voltage" on the battery, which is what charges it.