Large language models (LLMs) are increasingly used for automated data visualization, yet existing approaches often frame visualization generation as a single-step mapping from user query to figure or code, overlooking the iterative analytical reasoning process of expert analysts. We present InsightChain, a four-step visualization prompting pipeline (Explore-Focus-Test-Present) that emulates expert analytical workflows for complex data visualization, together with VG-COPRO, a novel automatic prompt optimization (APO) algorithm for jointly optimizing pipeline performance. To further bridge the evaluation gap for complex data visualization, we introduce the Insight Progression Metric (IPM), a rubric combining four text-based dimensions with a vision-based dimension. Experiments on public datasets demonstrate that InsightChain consistently outperforms competing prompting baselines. We further show that VG-COPRO is more effective than existing APO methods at optimizing chain-based visualization prompting strategies.