As a typical unsupervised machine learning task, clustering is always a hot research topic. Motivated by deep learning approaches, deep clustering has become prevalent in recent years, and achieves appealing performance. Most of current deep clustering methods focus on learning a new discriminative representation to enhance separability of original data, i.e., autoencoder, multilayer perceptrons and deep belief networks. However, the structure information which is very important in unsupervised learning, attracts little attention in previous deep feature representation learning clustering works. In this paper, a dynamic graph evolution based graph convolutional network (DGE-GCN) is introduced for clustering task, in which the data structural information and learned latent features are integrated into a unified network for deep clustering. Instead of using a fixed graph during the graph convolution process, we design a dynamic graph evolution strategy to refine the initial graph which could be not accurate. In addition, the latent representations learned by an autoencoder are embedded for refining the graph in a layer-wise manner. In such a manner, the latent features can help improve the graph structure, while the refined graph can in turn constrain the autoencoder to learn more discriminative features. In order to unify the graph convolutional network branch and autoencoder branch, a dual self-supervised mechanism is designed to guide the parameter learning of the whole network architecture. Comprehensive experiments demonstrate that the proposed network consistently performs better than several state-of-the-art methods on various benchmark datasets.