It's not clear from your schematic or your text -- you have a signal generator that is connected to a resistor and capacitor in series, and the resistor is grounded? Or is the capacitor grounded?
The following answers assume the resistor is grounded.
Method 1:
Get the RMS voltages of both the signal and the resistor voltage. Ignore phase, which means that you assume the capacitor is purely reactive. Then
V_c^2 + V_r^2 = V_s^2,
where V_c is the capacitor voltage, V_r is the resistor voltage and V_s is the signal generator voltage. This implies that V_c = sqrt(V_s^2 - V_r^2).
Now, the current is I = V_r/R, and the capacitive reactance is X_c = V_c/I, so you get X_c = R*sqrt(V_s^2 - V_r^2)/V_r. The capacitance can be found from the capacitive reactance and frequency, C = 1/(2*pi*X_c).
Method 2:
Assume that the capacitor isn't purely reactive. Measure the amplitude and phase of V_r respective to V_s, and solve for the capacitive impedance Z_c (note that you have to use complex arithmetic):
V_r Z_c = -----------. V_s - V_r
Z_c will, in general, be complex, so it will be Z_c = R_c - jX_c, where R_c is the capacitor's equivalent series resistance and X_c is the capacitive reactance. If Z_c is purely imaginary all is well and good. If Z_c has a significant resistive component then it is up to you to decide if this is measurement error or the fault of the capacitor, and whether it is best modeled as a series resistance, a parallel resistance, or something more complicated.