  We present a manifestly $N=2$ supersymmetric formulation of $N=2$ super-$W_3$
algebra (its classical version) in terms of the spin 1 and spin 2
supercurrents. Two closely related types of the Feigin-Fuchs representation for
these supercurrents are found: via two chiral spin $\frac{1}{2}$ superfields
generating $N=2$ extended $U(1)$ Kac-Moody algebras and via two free chiral
spin 0 superfields. We also construct a one-parameter family of $N=2$ super
Boussinesq equations for which $N=2$ super-$W_3$ provides the second
hamiltonian structure.
