Existing deep learning methods for clustering high-dimensional data perform feature selection and clustering separately, which can result in the exclusion of some important features for clustering. In this paper, we propose a method that performs deep clustering and feature selection simultaneously by inserting a concrete selector layer between the input layer and the first encoder layer of a modified autoencoder. The concrete selector layer performs feature selection, while the modified autoencoder performs clustering in the latent space by incorporating K-means loss and inter-cluster distances. The proposed method, called the K-concrete autoencoder, selects features important for clustering and uses only the selected features to learn K-means-friendly latent representations of the data. Moreover, we propose an extension of the K-concrete autoencoder to provide relative importance of each selected feature. We demonstrate the effectiveness of the proposed method using simulated and real datasets.