The advanced communication technology provides new monitoring and control strategies for smart grids. However, the application of information technology also increases the risk of malicious attacks. False data injection (FDI) is one kind of cyber attacks, which cannot be detected by bad data detection in state estimation. In this paper, a data-driven FDI attack detection framework of the smart grid with phasor measurement units (PMUs) is proposed. To enhance the detecting accuracy and efficiency, the multiple layer autoencoder algorithm is applied to abstract the hidden features of PMU measurements layer by layer in an unsupervised manner. Then, the features of the measurements and corresponding labels are taken as inputs to learn a softmax layer. Last, the autoencoder and softmax layer are stacked to form a FDI detection framework. The proposed method is applied on the IEEE 39-bus system, and the simulation results show that the FDI attacks can be detected with higher accuracy and computational efficiency compared with other artificial intelligence algorithms.