In this paper, we consider a class of noncooperative N-player finite-horizon linear-quadratic dynamic games with linear constraints. We introduce a constrained feedback information structure and provide necessary and sufficient conditions for the existence of a constrained feedback Nash equilibrium. For this class of games we show that the constrained feedback Nash equilibrium can be obtained from a feedback Nash equilibrium associated with an unconstrained multi-parametric linear-quadratic game where the parameters satisfy an equilibrium property. We show that this relation leads to a fixed-point interpretation. Further, under a few additional assumptions, we show that these fixed points can be obtained as solutions of a single large-scale linear-complementarity problem, thereby providing a method to compute the constrained feedback Nash equilibria. We illustrate our results with a numerical example.