Recent advances in 3D Gaussian Splatting (3DGS) have enabled fast, photorealistic rendering of dynamic 3D scenes, showing strong potential in immersive communication. However, in digital human encoding and transmission, the compression methods based on general 3DGS representations are limited by the lack of human priors, resulting in suboptimal bitrate efficiency and reconstruction quality at the decoder side, which hinders their application in streamable 3D avatar systems. We propose HGC-Avatar, a novel Hierarchical Gaussian Compression framework designed for efficient transmission and high-quality rendering of dynamic avatars. Our method disentangles the Gaussian representation into a structural layer, which maps poses to Gaussians via a StyleUNet-based generator, and a motion layer, which leverages the SMPL-X model to represent temporal pose variations compactly and semantically. This hierarchical design supports layer-wise compression, progressive decoding, and controllable rendering from diverse pose inputs such as video sequences or text. Since people are most concerned with facial realism, we incorporate a facial attention mechanism during StyleUNet training to preserve identity and expression details under low-bitrate constraints. Experimental results demonstrate that HGC-Avatar provides a streamable solution for rapid 3D avatar rendering, while significantly outperforming prior methods in both visual quality and compression efficiency.