We propose a kinetic chain growth algorithm for coarse-grained (CG) simulations in this work. By defining the reaction probability, it delivers a description of consecutive polymerization process. This algorithm is validated by modeling the process of individual styrene monomers polymerizing into polystyrene chains, which is proved to correctly reproduce the properties of polymers in experiments. By bridging the relationship between the generic chain growth process in CG simulations and the chemical details, the impediment to reaction can be reflected. Regarding to the kinetics, it models a polymerization process with an Arrhenius-type reaction rate coefficient. Moreover, this algorithm can model both the gradual and jump processes of the bond formation, thus it readily encompasses several kinds of previous CG models of chain growth. With conducting smooth simulations, this algorithm can be potentially applied to describe the variable macroscopic features of polymers with the process of polymerization. The algorithm details and techniques are introduced in this article.