Although Large Language Models (LLMs) excel in many tasks, their application to Speech-to-Speech Translation (S2ST) is underexplored and hindered by data scarcity. To bridge this gap, we propose PROST-LLM (PROgressive Speech-to-speech Translation) to enhance the S2ST capabilities in LLMs progressively. First, we fine-tune the LLMs with the CVSS corpus, employing designed tri-task learning and chain of modality methods to boost the initial performance. Then, leveraging the fine-tuned model, we generate preference pairs through self-sampling and back-translation without human evaluation. Finally, these preference pairs are used for preference optimization to enhance the model's S2ST capability further. Extensive experiments confirm the effectiveness of our proposed PROST-LLM in improving the S2ST capability of LLMs.