you can check the voltage of a battery with a VOM, but it won't necessarily tell you the health/capacity of the battery. A dying battery's voltage might look fine just after charging, but it might quickly lose it's charge after running under load.
although the dock connector pinout is the same across all models, I don't believe you can check the battery voltage/status from the dock connector, instead open up the iPod and remove the battery.
are you finding poor battery life with your iPod? if you charge it fully and are noticing a decrease in battery life compared to when it was first new, it's quite possible your battery is dying - they don't last forever, unfortunately. However, they're easy and inexpensive to replace