  We study the superconducting transition temperature and the electronic
properties of the metallic phase of $\kappa$-type (BEDT-TTF)$_2$X which shows
unconventional properties in experiments, on the basis of the third order
perturbation theory for a simple effective Hubbard model of a nearly triangular
lattice. Appropriate transition temperatures and $d_{x^2-y^2}$ symmetry of the
gap function are obtained in good agreement with experimental results. We also
calculate the transition temperature by the fluctuation-exchange
approximation(FLEX) in order to compare the two approaches; FLEX gives higher
transition temperatures rather than the perturbation approach. However, it is
also found that the vertex corrections, which are ignored in FLEX, have a
crucial effect on $T_{\rm c}$ for strongly frustrated systems. The density of
states and the normal self-energy calculated in this perturbation scheme show
the nature of the conventional Fermi liquid near the Mott-insulator. Thus, our
perturbation approach is applicable to the conventional metallic phase of this
compound, while it cannot explain the (pseudo-)spin gap phenomenon which
signals the non-Fermi liquid.
