Fuzzy clustering is a classical approach to provide the soft partition of data. Although its enhancements have been intensively explored, fuzzy clustering still suffers from the difficulties in handling real high-dimensional data with complex latent distribution. To solve the problem, this article proposes a deep fuzzy clustering method by representing the data in a feature space produced by the deep neural network. From the perspective of representation learning, three constraints or objectives are imposed to the neural network to enhance the clustering-friendly representation. At first, as a good representation of data, the mapped data in the new feature space should support the reconstruction of original data. So, the autoencoder architecture is applied to ensure that the original data can be recovered by decoding the encoded representation with another neural network. Second, to solve the clustering problem efficiently, the intracluster compactness and the intercluster separability are to be minimized and maximized, respectively, in the new feature space. At last, considering that the data in the same class should be close to each other, the affinities between new representations are tuned in accordance with the discriminative information. Altogether, we design a graph-regularized deep normalized fuzzy compactness and separation clustering model to conduct representation learning and soft clustering simultaneously. The learning algorithm based on stochastic gradient descent is proposed to the model, and the comparative studies with baseline clustering algorithms on real-world data illustrate the superiority of the proposal.