Single-cell RNA sequencing (scRNA-seq) technology provides high-throughput gene expression data to study the cellular heterogeneity and dynamics of complex organisms. Graph neural networks (GNNs) have been widely used for automatic cell type classification, which is a fundamental problem to solve in scRNA-seq analysis. However, existing methods do not sufficiently exploit both gene-gene and cell-cell relationships, and thus the true potential of GNNs is not realized. In this work, we propose a bilevel graph representation learning method, named scBiGNN, to simultaneously mine the relationships at both gene and cell levels for more accurate single-cell classification. Specifically, scBiGNN comprises two GNN modules to identify cell types. A gene-level GNN is established to adaptively learn gene-gene interactions and cell representations via the self-attention mechanism, and a cell-level GNN builds on the cell-cell graph that is constructed from the cell representations generated by the gene-level GNN. To tackle the scalability issue for processing a large number of cells, scBiGNN adopts an Expectation Maximization (EM) framework in which the two modules are alternately trained via the E-step and M-step to learn from each other. Through this interaction, the gene- and cell-level structural information is integrated to gradually enhance the classification performance of both GNN modules. Experiments on benchmark datasets demonstrate that our scBiGNN outperforms a variety of existing methods for cell type classification from scRNA-seq data.