From a general point of view there is not so much formal difference between the permittivity of a dielectric and conductance of a metal. For the latter, a crude classical model is that for a quasi-free gas of conduction electrons, moving in the external electric field and subject to friction from collisions. The equation of motion for such a conduction electron reads (in non-relativistic approximation, which is sufficient for any practical purposes)
m \dot{\vec{v}}=-m \gamma \vec{v}+q\vec{E}.
Neglecting the spatial variation of \vec{E} along the relevant distances of the electron's motion, we find by finding the retarded Green's function of the operator \partial_t+\gamma
\vec{v}(t)=\int_{-\infty}^{\infty} \mathrm{d} t' \Theta(t-t') \exp[-\gamma(t-t')]\frac{q}{m} \vec{E}(t').
The current is given by
\vec{j}(t)=n q \vec{v} = \frac{n q^2}{m}\mathrm{d} t' \Theta(t-t') \exp[-\gamma(t-t')]\frac{q}{m} \vec{E}(t'),
where n is the conduction-electron density, which we also consider as spatially homogeneous.
Writing the electric field as Fourier transform,
\vec{E}(t,\vec{x})=\int_{-\infty}^{\infty} \frac{\mathrm{d} \omega}{2 \pi} \exp(-\mathrm{i} \omega t) \tilde{\vec{E}}(\omega,\vec{x})
and also
\vec{j}(t,\vec{x})=\int_{-\infty}^{\infty} \frac{\mathrm{d} \omega}{2 \pi} exp(-\mathrm{i} \omega t) \tilde{\vec{j}}(\omega,\vec{x}),
we find in the frequency domain
\tilde{\vec{j}}(\omega,\vec{x})=\sigma(\omega) \tilde{\vec{E}}(\omega,\vec{x})
with
\sigma(\omega)=\frac{n q^2}{m} \frac{1}{\gamma-\mathrm{i} \omega}.
The only difference in the case of a non-conducting dieelectric is that here all the electrons are bound and in linear-response theory can be described as moving in a harmonic potential with frequency \omega_0 and some friction coefficient\gamma. The only difference here is that we express the response to a (weak) electric field in terms of the polarization
\vec{P}=n q \vec{x}
In Fourier space the Green's function gives the electric susceptibility,
\chi_e(\omega)=\frac{n q^2}{m} \frac{1}{\omega_0^2-\mathrm{i} \gamma \omega-\omega^2},
leading to
\tilde{\vec{P}}(\omega,\vec{x})=\chi_e(\omega) \tilde{\vec{E}}(\omega,\vec{x}).
In Heaviside-Lorentz units one thus has
\tilde{\vec{D}}=\tilde{\vec{E}}+\tilde{\vec{P}}=(1+\chi_e) \tilde{\vec{E}},
and thus
\epsilon(\omega)=1+\chi_e(\omega).
In a real material you have several eigen frequencies, corresponding to the different quantum-theoretical bound states of the electrons to their ions, and also in a conductor you usually have some bound electrons, so that \sigma and \chi_e are given by the sum of the corresponding terms.
That there is not so much difference in the two cases, because on a microscopic level the total current is given by the conduction current and the current due to the motion of the bound electrons. The latter obviously is given by
\vec{j}_{\text{bound}}=\partial_t \vec{P}
or, in frequency space,
\tilde{\vec{j}}_{\text{bound}}=-\mathrm{i} \omega \tilde{\vec{P}}.
Thus one finds the conductivity for \omega_0=0 for the dielectric case via this relation
\sigma=\left .-\mathrm{i} \chi_e \right|_{\omega_0=0}.
