What is the process by which a cell duplicates its chromosomes and splits into two cells?

Cell division. The whole process involves 3 main stages namely interphase, mitosis and cytokinesis. It would, however be wrong to say that its mitosis only because mitosis does not involve the duplication of chromosomes. Duplication of chromosomes occur during the interphase while the splitting of cells occur during Telophase of mitosis and Cytokinesis. Therefore, answer to this has to be generalised to be just cell division.