In this paper, we consider the convergence analysis of data-driven mathematical programs with distributionally robust chance constraints (MPDRCC) under weaker conditions without continuity assumption of distributionally robust probability functions. Moreover, combining with the data-driven approximation, we propose a DC approximation method to MPDRCC without some special tractable structures. We also give the convergence analysis of the DC approximation method without continuity assumption of distributionally robust probability functions and apply a recent DC algorithm to solve them. The numerical tests verify the theoretical results and show the effectiveness of the data-driven approximated DC approximation method.