In this paper, we consider the biharmonic elliptic systems of the form $${{Delta}^2u=F_u(u,v)+{lambda}{mid}u{mid}^{q-2}u,;x{in}{Omega},\{Delta}^2v=F_v(u,v)+{delta}{mid}v{mid}^{q-2}v,;x{in}{Omega},\u=frac{{partial}u}{{partial}n}=0,; v=frac{{partial}v}{{partial}n}=0,;x{in}{partial}{Omega},$$, where ${Omega}{subset}mathbb{R}^N$ is a bounded domain with smooth boundary ${partial}{Omega}$, ${Delta}^2$ is the biharmonic operator, $N{geq}5$, $2{leq}q$ < $2^*$, $2^*=frac{2N}{N-4}$ denotes the critical Sobolev exponent, $F{in}C^1(mathbb{R}^2,mathbb{R}^+)$ is homogeneous function of degree $2^*$. By using the variational methods and the Ljusternik-Schnirelmann theory, we obtain multiplicity result of nontrivial solutions under certain hypotheses on ${lambda}$ and ${delta}$.