A time-dependent density-functional theory for systems in periodic external potentials in time is formulated on the assumption of the existence of the Floquet states from the quasienergy viewpoint. Coupling strength integration, which connects a noninteracting system with an interacting system, is introduced by using the time-dependent Hellmann-Feynman theorem. Coupled perturbed time-dependent Kohn-Sham equations are derived from the variational condition to the quasienergy functional with respect to parameters. Explicit expressions for frequency-dependent polarizability and first hyperpolarizability are given by the quasienergy derivative method. Excitation energies and transition moments are defined from poles and residues of frequency-dependent polarizabilities, respectively. In contrast to the previous theory, our formulation has the following three advantages: (1) The time-dependent exchange-correlation potential is defined by the functional derivative of the exchange-correlation quasienergy. (2) The formal expression for frequency-dependent polarizability, which corresponds to the exact sumover-states expression, can be obtained. (3) Explicit expressions for response properties which satisfy the 2n+1 rule can be automatically obtained.