Spectral clustering is one of the most popular modern clustering algorithms. It is easy to implement, can be solved efficiently, and very often outperforms other traditional clustering algorithms such as k-means. However, spectral clustering could be insufficient when dealing with most datasets having complex statistical properties, and it requires users to specify the number k of clusters and a good distance metric to construct the similarity graph. To address these problems, in this article, we propose an approach to extending spectral clustering with deep embedding, cluster estimation, and metric learning. First, we generate the deep embedding via learning a deep autoencoder, which transforms the raw data into their lower dimensional representations suitable for clustering. Second, we provide an effective method to estimate the number of clusters by learning a softmax autoencoder from the deep embedding. Third, we construct a more powerful similarity graph by learning a distance metric from the embedding using a Siamese network. Finally, we conduct an extensive experimental study on image and text datasets, which verifies the effectiveness and efficiency of our approach.