Research on the community structure of networks is beneficial for understanding the structure of networks, analyzing their characteristics and discovering the rules hidden in these networks. To address issues from previous community mining algorithms, such as the low rate of convergence and high time complexity, this study proposes an improved community structure discovery algorithm named CPMK-Means algorithm. The main idea of this algorithm can be summarised as follows. The clique percolation method (CPM) algorithm generates the maximum number of cliques by combining depth-first search with breadth-first search so that the number of cluster centres is determined. Then, the k centres are selected based on the principle of the maximum degree of centres and minimum similarity between different centres. Afterwards, nodes in the network are assigned to the communities formed by the k centres, and the iterations are performed repeatedly until the centres become stable. Finally, the overlapping communities are merged. Experiments are carried out on standard data sets Football and Collins to evaluate the performance of the CPMK-Means algorithm. Results indicate that the CPMK-Means algorithm can achieve better community mining and higher execution efficiency compared with other algorithms. Furthermore, it is superior to other algorithms in terms of precision, recall, accuracy, F-measure and separation.