  We construct realizations of the generators of the $\kappa$-Minkowski space
and $\kappa$-Poincar\'{e} algebra as formal power series in the $h$-adic
extension of the Weyl algebra. The Hopf algebra structure of the
$\kappa$-Poincar\'{e} algebra related to different realizations is given. We
construct realizations of the exterior derivative and one-forms, and define a
differential calculus on $\kappa$-Minkowski space which is compatible with the
action of the Lorentz algebra. In contrast to the conventional bicovariant
calculus, the space of one-forms has the same dimension as the
$\kappa$-Minkowski space.
