How to measure the saturation current of a diode
Two measurements should be enough for the basic Shockley model.


Measure current with fairly large reverse bias, to give you Is. (for example, -20V for a 1N4148)
Measure forward voltage with fairly large forward current (for example, 20mA for a 1N4148) and calculate the emission coefficient: 

\$n = \frac{V_F}{V_T \cdot ln(I/I_S)}\$


Where 

Vf is measured forward voltage

I is the test current 

Is is the saturation current from step 1

Vt is the thermal voltage calculated from kT/q where T is the junction temperature 
   in Kelvin, q is the charge of an electron and k is the Boltzmann constant. 

Very small diodes (or larger diodes at high current) will have a significant resistive term (not modeled by Shockley) that may become significant, in which case you can plot n vs. I over a range to eliminate that effect (or make at least one more measurement and eliminate it mathematically).