Well you are fundementally working with a 10bit ADC converter that AVR datasheet rates as:
• ± 2 LSB Absolute Accuracy
So 8 bits of accuracy you can count on at all times? While I have no doubt that you can learn a lot in trying to design and build such an arduino project the possibility that you could even match the performance of a cheap $10 DMM is very low in my opinion. If I was to pursue such a project I would at least consider utilizing an external I2C ADC module with much better fundamental accuracy capabilities, such as:
Lefty