What measures of the dielectric constant mark whether the solution is ionic, polar and nonpolar?
Just compare the dielectric constant of known substances to non-polar, polar, metallic designations. For example, the dielectric constant of water is around 80.0 and water is polar. H2 has a very low constant value near 2 I believe and it is very non-polar. Metals have very large epsilon values moreover.  Rule of thumb, the larger the dielectric constant, the more polar the substance or material. In math terms, the potential felt between 2 charges is V= (q1 q2) / (4pi epsilon r) where q1 and q2 are different charges separated by r. The higher epsilon, the smaller V becomes. How charges interact with each other hence depends on the dielectric constant of the medium or material.