In this paper we study and give optimal estimates for the Dirichlet problem for the biharmonic operator ${\Delta }^2$, on an arbitrary bounded Lipschitz domain $D$ in ${\mathbf{R}}^n$. We establish existence and uniqueness results when the boundary values have first derivatives in $L^2\left(\partial D\right)$, and the normal derivative is in $L^2\left(\partial D\right)$. The resulting solution $u$ takes the boundary values in the sense of non-tangential convergence, and the non-tangential maximal function of $\nabla u$ is shown to be in $L^2\left(\partial D\right)$.