In this work we will consider a class of second order perturbed Hamiltonian systems with a superquadratic growth condition on a time periodic potential and a small aperiodic forcing term. To get an almost homoclinic solution we approximate the original system by time periodic ones with larger and larger time periods. These approximative systems admit periodic solutions, and an almost homoclinic solution for the original system is obtained from them by passing to the limit, as the periods go to infinity, in the topology of almost uniformly convergence of functions and derivatives up till the order 2. Our aim is to show the existence of two different approximative sequences of periodic solutions: one of mountain pass type and the second of local minima.