Temporal Knowledge Graph Completion (TKGC) aims to infer missing facts by modeling historical events and latent temporal dependencies in Temporal Knowledge Graphs (TKGs). Recently, TKGC methods that integrate graph embeddings into Large Language Models (LLMs) have shown great promise by leveraging the structural information of TKGs together with the powerful reasoning capabilities of LLMs. However, these embedding-based methods are limited by suboptimal graph representations due to noise and long-tail issues in real-world scenarios, and insufficient cross-modal alignment between graph and language, hindering LLMs' ability to fully capture the temporal and structural information of TKGs. To address these issues, we propose TGCA-LLM, a novel embedding-based framework for TKGC. Specifically, TGCA-LLM first employs time-aware contrastive learning to align fact texts with graph structures in the temporal dimension, generating robust graph embeddings and establishing initial cross-modal alignment. Then, through a two-stage tuning process, it enables LLMs to gradually acquire structural and temporal knowledge from graph embeddings while enhancing their cross-modal reasoning capabilities in TKGC. Extensive experiments on three widely used real-world benchmarks demonstrate that TGCA-LLM outperforms state-of-the-art (SOTA) baselines by at least 8.7% MRR, highlighting its effectiveness.