Multivariate time series classification (MTSC) refers to assigning the correct classification label to each multivariate time series sample, which can be effectively addressed by popular deep learning technology. Recent approaches integrate graph neural networks (GNNs) with diverse temporal modeling frameworks to capture spatial and temporal dynamics. However, existing GNN-based MTSC methods are limited by the structural deficiencies of deep GNNs, making it challenging to enhance the sample representation by extracting relationships in the MTS sample graph to improve the classification performance. To solve this problem, we propose a novel MTSC model based on the deep graph convolution network based on a variational autoencoder (VAE-GCN). Specifically, to address the feature homogenization problem of MTS samples caused by the over-smoothing phenomenon in deep GCNs, we propose a deep GCN based on generalized sample representation injection to extract the latent relationships among MTS samples more effectively. Further, we design a generalized sample representation extraction mechanism based on the variational autoencoder to fuse representative MTS sample category features with higher-order relationships of MTS samples, which alleviates the tendency of MTS sample features converging to the mean during the feature propagation process of our model. Experimental results compared with the ten state-of-the-art classification baselines on 11 real datasets significantly demonstrate the superiority of our approach.