We study phase structure and critical point of finite-temperature QCD with heavy quarks applying the hopping parameter expansion (HPE). We first study finite-size effects on the critical point on $N_t=4$ lattices with large spatial volumes taking the LO and NLO effects of the HPE, and find that the critical scaling of the Z(2) universality class expected around the critical point of two-flavor QCD is realized when the aspect ratio of the lattice is larger than about 9. This enables us to determine the critical point in the thermodynamic limit with high precisions. By a study of the convergence of the HPE, we confirm that the result of the critical point with the low orders of the HPE is reliable for $N_t=4$, while we need to incorporate higher order effects for $N_t \ge 6$. To extend the study to $N_t \ge 6$ lattices, we then develop a method to take the effects of higher-order terms of the HPE up to a sufficiently high order. We report on the status of our simulations on $N_t \ge 6$ lattices adopting the new method.