all is Ok except from two-electrode cell. To perform CVA experiment you need reference electrode. Mercury oxide/mercury electrode is preferable in alkaline medium. AgCl/Ag electrode works quite well too.
in a symmetric activated carbon based capacitor, may be it is not a problem to get a CV in this voltage range. But you have not mentioned for which purpose you want a CV and what you want to investigate.
Two electrode cell setup will tell you about the overall capacitance/performance of a cell
while three electrode setup will tell about the processes going on at each electrode e.g., oxidation and reduction limits of the system, and the stability window or potential range information
As mentioned above, it is very important to know what you want from your CV in order to know what the best way to do it is. It is possible to do CVs in both 2-electrode and 3-electrode configurations, and carbon should be fine in that particular electrolyte. However, not only is the information you will obtain different depending on your configuration, but as the actual potential at the working electrode will change, the processes that will occur in your system can vary greatly.... so unless you provide us with a little more information as to what you actually want, it is difficult to make any useful suggestions.
The two-electrode system was setup for the purpose of obtaining capacitance info of the material. If the CV was done in the range of -1V to 1V, is it ok to report the specific capacitance in that range, according to CV data?
Are you sure that you want the same material as the anode and cathode elecctrodes for a capacitor? For a better understanding of your syestem, you may perform the experiments in a two-step process. Step 1: you could use a three-electrode system to check the electrochemical behaviour of activated carbon as anothode or cathode, to find out the double-layer potenial window. Step 2: use the two-electrode system as you stated.
I think that, as Qamar mentioned, if you want the overall capacitance of the cell, then yes, you can do so in the system you described, no problem!
If you happen to want to know the capacitance of just an activated carbon electrode, then I'd suggest replacing one of the electrodes by a different material, preferably a simple metal plate.
I am afraid that you will not be able to polarize the capacitor (two-electrode setup) with activated carbon in KOH between -1 V to +1 V. The potential window of KOH on an activated carbon electrode is maximum 1 V that means starting from 0 V or open circuit voltage you can go up to +1 V.
For capacitance to report, I would advise you to do galvanostatic charge discharge, and then from discharge curve, calculate the capacitance. for both CV and galvanostatic charge discharge, polarize the two electrode cell from 0 V to +1 V.
However if you want to realize a three electrode experiment, as emphasized by Daniela and Qijin, just add a reference electrode to your system (Ag/AgCl mentioned by Georgii). In this case make one activated carbon electrode as working electrode, then you can polarize in the range from -1 V to lets say +1 V but I am sure that you will see an oxidation peak at around +0.1 V vs NHE because KOH potential window will be limited by that peak.
First of all, I have serious objection on this CV. In a two-electrode cell which represents a real capacitor why you want to polarize it to negative voltage? this energy is useless from practical point of view. In a two-electrode setup always polarize your cell starting from open circuit voltage which is around 0 V in the positive voltage direction.
Only in a three-electrode cell where one of your electrode is a working electrode, you can polarize between whatever potential range, to check the stability of electrode material or that of electrolyte.
Now if you look at this picture of two-electrode that you posted, and consider polarizing your capacitor from 0 V, you have good charge propagation and hence good capacitance. But moving in positive voltage direction, you start to have an upward increase in current at about 0.7 V that means you should not polarize your capacitor beyond this voltage unless there is good reversibility. But in my opinion, using KOH electrolyte you can polarize up to +0.8 V or maximum up to +1 V. The reason is very high pH value of KOH that limits the operation of positive electrode.
You have to understand the difference between three electrode and two electrode setup and the difference between potential and voltage.
Now if you want to have some aqueous electrolyte with higher stability window, I would suggest you to use 1 mol/L solution of LiNO3 or Li2SO4. With these electrolytes you can polarize your two electrode system up to 1.6 V. This is because of near neutral pH values of these salt solution electrolytes.
If you have further question, do not hesitate to ask