Most chargers don't have a programmable microcontroller or software... But yes, it's feasible.
to build a lipo charger with an accuracy of down to 0.001v per cell?
No... The Arduino has a 10 bit ADC (analog-to-digital converter). That gives you 1024 "steps" for 5mV resolution on the 5V range or 1mV resolution on the optional 1.1V reference (too low for your battery).
If you are charging in series you'll need a voltage divider, or multiple voltage dividers, to keep the voltage below the 5V maximum and your resolution will be reduced proportionally.