I'm not really sure how the INA125 pins 4 & 13 really work. Please could you explain it to me.
It sets the reference voltage...its effect can be seen in the formula below.
I think the main difficulty here in being able to use the full range of Arduino's ADC, is that there is no -5V supply, so you need to make single supply connections as in figure 6. In this diagram, they are connecting pins 4 & 14 for 2.5V reference. Note that (load -) is also Vref, so the maximum linear range (I think) would be from code 512 to 1023.
So, by lowering Vref to 1.24V you could achieve maximum linear range of code 256 to 1023.
With a -5V supply and figure 1 diagram, you could achieve full range (0 to 1023).
The formula here tells all: