A capacitor is just two conductors (usually metal plates or metal films) separated by an insulator (air, mica, ceramic or various plastics.)
A reverse biased diode does not conduct because there is a region, the depletion region, which has few mobile charge carriers (electrons or holes) and behaves like an insulator. So you have a sandwich of an insulating region between two conductors, the anode and cathode, which makes it behave like a capacitor.*
Having got your varicap diode, I think you could measure its capacitance at various voltages by building an oscillator and measuring the frequency of oscillation. You would need to calibrate the circuit by using known values of capacitance in place of the varicap.
You might not need to measure the frequency, so long as you can detect and identify specific frequencies, say with a radio receiver, then adjust the varicap to match the frequency generated by a standard capacitor.
So make your oscillator circuit with a known capacitor about the value you expect for the varicap (err on the low side.) Design it to oscillate at a frequency near the middle of the range your radio can receive.
Tune the radio to the frequency you get, then leave it there.
Replace the fixed capacitor with the varicap circuit and adjust until you hear it on the radio the same as before.
Repeat with other values of known capacitor. (Or maybe use a known variable capacitor in different positions.)
*(Incidentally, this is one of the major limitations on switching transistors: the capacitances between electrodes have to be charged and discharged every time the transistor switches off and on.)