Conductivity is the inverse of resistance. One way to measure it is to apply a small AC voltage across the sample with a known resistor in series. Measure the voltage across the electrodes and get current by Ohm's law.
Using AC prevents building up deposits on electrodes that will throw results out. If you can arrange to use a cuvette with an interior area of a cm square or some known multiple, then the data are more easily presented in mhos/cm sq.
Remember though that your results are going to be temperature dependent, so that if you do have not have a set temperature for your readings your results are going to be incorrect.
Any air is going to throw off your conductivity readings as well.
I recall a physics experiment where we used the output from a signal generator, and measured the AC current across the resistor at various frequencies.
A plot of the resistance against frequency was then extrapolated back to zero frequency to obtain the DC value.