As high-frame-rate synthetic aperture radar (SAR) has the ability to form continuous SAR images and dynamically monitor the ground areas of interest, it has attracted more and more attention nowadays. In practical application, the enormous data in high-frame-rate SAR system to obtain the multi-frame images brings big challenges to its transmission, storage, and processing. In order to solve this problem, there are many recent papers on formulating the high-frame-rate SAR imaging problem into a low-rank tensor recovery problem, and correspondingly, the sampling amount of the high-frame-rate SAR data can be largely reduced. However, existing algorithms to solve the low-rank tensor recovery problem in high-frame-rate SAR application still suffer from large computational cost. Under the above inspiration, this paper proposes a deep neural network architecture for high-frame-rate SAR imaging, i.e., Tensor Alternating Direction Method of Multiplier Network (TADMM-Net), which is more computational efficient in imaging procedure. Specifically, we formulate the high-frame-rate SAR imaging processing into a low-tubal-rank tensor recovery problem. We solve the low-tubal-rank tensor recovery problem using a tensor Alternating Direction Method of Multiplier (ADMM) algorithm and then design a new deep neural network architecture by applying algorithm unfolding techniques to the underlying low-rank tensor recovery problem. The proposed TADMM-Net approach shifts the computational burden from the testing phase to the training phase, and the practical processing time can be extremely decreased compared with the existing algorithms for the low-rank tensor recovery problem in high-frame-rate SAR imaging application. It also offers various advantages over the existing high-frame-rate SAR imaging algorithms including higher performance in the case with low sampling amount, lower storage complexity and no requirement for hand-craft hyper-parameters adjustment. The methodology was tested on high-frame-rate SAR data. These tests show that the proposed architecture outperforms other state-of-the-art methods in high-frame-rate SAR imaging applications.