This paper presents some constrained ${C}^{0}$ finite element approximation methods for the biharmonic problem, which include the ${C}^{0}$ symmetric interior penalty method, the ${C}^{0}$ nonsymmetric interior penalty method, and the ${C}^{0}$ nonsymmetric superpenalty method. In the finite element spaces, the ${C}^{1}$ continuity across the interelement boundaries is obtained weakly by the constrained condition. For the ${C}^{0}$ symmetric interior penalty method, the optimal error estimates in the broken ${H}^{2}$ norm and in the ${L}^{2}$ norm are derived. However, for the ${C}^{0}$ nonsymmetric interior penalty method, the error estimate in the broken ${H}^{2}$ norm is optimal and the error estimate in the ${L}^{2}$ norm is suboptimal because of the lack of adjoint consistency. To obtain the optimal ${L}^{2}$ error estimate, the ${C}^{0}$ nonsymmetric superpenalty method is introduced and the optimal ${L}^{2}$ error estimate is derived.