What's the relationship between aileron deflection and roll moment?
Using a linear theory, the roll moment coefficient can be expressed as:

$$
C_l =C_{l0}+C_{l\beta}\beta+C_{l\delta_{\alpha}}\delta_{\alpha}+C_{l\delta_{r}}\delta_{r}
$$

, where:
$\beta$ : slip angle
$\delta_{\alpha}$ : aileron deflection
$\delta_{r}$ : rudder deflection

This doesn´t really say much, as we are simply defining this coefficient in a way that is convenient. This equation just says that $C_l$ is equal to a linear composition of the effects of assymetry, slip angle, and aileron and ruder deflections, each weighed by a stability derivative.
Some of these stability derivatives have proper names or are trivially discarded:


$C_{l0}=(C_l)_{\beta=\delta_{\alpha}=\delta_r=0}=0$ , for symmetric aircraft without engine torques 
$C_{l\beta}$ is the dihedral effect, or the effect of slip on roll. It is the sum of contributions from the wing-body assembly, the HTP and the VTP. The wing-body effect can be further split into: the geometric dihedral of the wing, wing sweep and vertical position (high or low).

$C_{l\beta} =(C_{l\beta})_{WB}+(C_{l\beta})_{HTP}+(C_{l\beta})_{VTP}$

$C_{l\beta}=-\frac{a_w \Gamma}{4}-\frac{C_L}{4}sin(2\Lambda)+(C_{l\beta})_{\text{wing height}}+(C_{l\beta})_{HTP}-a_v\eta_v\frac{S_v}{S}(1+\frac{\partial\sigma}{\partial\beta})\frac{h_v}{b}$

$(C_{l\beta})_{HTP}$ is obtained the same way as the WB contribution, although with different adimensionalizers.
For general reference, wingtips above the root are stabilizing, positive sweep is stabilizing and the wing height effect depends on the fuselage fraction ahead and behind the wing, but usually a high wing should be stabilizing.

$C_{l\delta_{\alpha}}$ is the roll control power or the effect of ailerons on roll. Is is calculated directly from the Prandtl wing theory by considering the impact of aileron deflection on the lift distribution
$C_{l\delta_{r}}$ is the effect of rudder on roll

$C_{l\delta_{r}}=C_{Y\delta_r}\frac{h_v}{b}=-a_v\eta_v\frac{S_vh_v}{Sh}\tau_r$




Your original question asked for the roll rate, and Peter Kämpf does indeed provide an answer, but it is probably no more useful to your particular need than my very first equation.

If you are interested in the roll rate, you will need to solve the lateral-directional system. You can simplify it somewhat by allowing fixed controls and level fight, thus obtaining a stability cuartic. This has, for most configurations, two real and two complex conjugate solutions,  corresponding to the three lateral-directional eigenmodes: Roll Subsidience (large negative real root), Spiral Divergence (small real root, sometimes even >0) and Dutch Roll (complex conjugate root). Since the large negative real eigenvalue has a much greater modulus than the rest, its mode (Roll Subsidience) will dominate for short periods, and you can obtain a good approximation of the roll transfer function by doing $\Delta\beta=\Delta\hat{r}=\Delta\delta_r=0$ and $C_{l\hat{\dot{\delta}}_\alpha}\cong0$:

$$
G_{\hat{p}\delta_\alpha}=\frac{\Delta\hat{p}(s)}{\Delta\delta_\alpha(s)}=\frac{C_{l\delta_\alpha}}{\hat{I_x}}\frac{1}{s-(C_{l\hat{p}}/\hat{I_x})}
$$

...in the Laplace plane. Keep in mind the parameters are non-dimensional.