Typhoons are some of the most serious natural disasters, and the key to disaster prevention and mitigation is typhoon level classification. How to better use data of satellite cloud pictures to achieve accurate classification of typhoon levels has become one of classification the hot issues in current studies. A new framework of deep learning neural network, Graph Convolutional–Long Short-Term Memory Network (GC–LSTM), is proposed, which is based on the data of satellite cloud pictures of the Himawari-8 satellite in 2010–2019. The Graph Convolutional Network (GCN) is used to process the irregular spatial structure of satellite cloud pictures effectively, and the Long Short-Term Memory (LSTM) network is utilized to learn the characteristics of satellite cloud pictures over time. Moreover, to verify the effectiveness and accuracy of the model, the prediction effect and model stability are compared with other models. The results show that: the algorithm performance of this model is better than other prediction models; the prediction accuracy rate of typhoon level classification reaches 92.35%, and the prediction accuracy of typhoons and super typhoons reaches 95.12%. The model can accurately identify typhoon eye and spiral cloud belt, and the prediction results are always kept in the minimum range compared with the actual results, which proves that the GC–LSTM model has stronger stability. The model can accurately identify the levels of different typhoons according to the satellite cloud pictures. In summary, the results can provide a theoretical basis for the related research of typhoon level classification.