How to estimate the dielectric constant of a liquid?
Yes, such relations exist. 

Notably there are two general theories for polar and non-polar fluids that connect the polarizability $P$ with the dielectric constant $\epsilon$ as reviewed in this article by Harvey and Lennon, International Journal of Thermophysics, Vol. 26, No. 1, January 2005.

The Clausius-Mosotti equation for non-polar liquids is:
$$P = \frac{\epsilon-1}{\epsilon +2}$$

Whereas an equation developed by Kirkwood for polar liquids is:
$$ P = \frac{(\epsilon-1)(2\epsilon+1)}{9 \epsilon}$$

The article then continues to explain that you can estimate the polarizability from theoretical relations involving the molar density $\rho$ and the dipole moment $\mu$, but in your case the two equations above are sufficient since you assume that you know $P$.