The reason why you are getting an offset voltage on the output of the AD623 is that the PNP transistors on input of the instrumentation amplifier do not have a path for the input bias current to flow, which results in an offset voltage at the output.
lladnar23 is right. You could just bridge the cap with a 10M resistor, which would approximate the leakage current in a realistic capacitor, or you could do the best solution, which is to add an explicit path for the DC input bias current.
This solution is shown on Figure 47c on page 14 on the AD623 DATASHEET. Please read the datasheet.
Is there a particular reason why you are using an instrumentation amp in a single-ended measurement (ct_1.jpg)? It really would be cheaper to do this measurement with an op-amp, and you could use a nice JFET input opamp and blissfully ignore input bias current issues.
By the way, is this some sort of school project? I noticed that there is another post about AD623 issues with a very similar circuit.
-elcap