In this paper, crystal growth instability of diamond was studied in a Fe-Ni-C system at high temperature-high pressure (HPHT). As any other crystal grown from solution, the flat or smooth growth interface of the diamond crystal is highly sensitive to growth conditions. The growth front interface should be of great importance to understand the diamond growth process. The presence of cellular growth interface by transmission electron microscopy indicated that there existed a narrow constitutional supercooling zone in front of the growth interface. Several parallel layers with cellular interface by TEM directly suggested that the diamond grows from the solution of carbon in the molten catalyst layer by layer, which is in accordance with the result obtained by scanning electron microscopy in this paper.Impurities are trapped by rapidly advancing growth layers during the diamond growth and they impose a great effect on the growth front stability. As the growth front interface approaches the impurity particle to a distance of about 10-5～10-7 cm, appreciable molecular forces begin to operate between them, and the impurity particle is trapped as the growth rate reaches a critical value. As a result, the driving force for crystallization under the impurity particles becomes smaller, the front buckles under the particle. An impurity naturally reduces the growth rate to a different extent.