Recent advances in text-to-speech (TTS) synthesis, particularly those leveraging large language models (LLMs), have significantly improved expressiveness and naturalness. However, generating human-like, interactive dialogue speech remains challenging. Current systems face limitations due to the scarcity of dual-track data and difficulties in achieving naturalness, contextual coherence, and interactional dynamics, such as turn-taking, overlapping speech, and speaker consistency, in multi-turn conversations. To address these challenges, we propose DialoSpeech, a dual-track architecture combining a large language model with Chunked Flow Matching for expressive, human-like dialogue speech synthesis. DialoSpeech generates natural multi-turn conversations with coherent speaker turns and natural overlaps, supporting both Chinese and English and cross-lingual speech synthesis. We introduce a data processing pipeline to construct dual-track dialogue datasets, facilitating scalable training and experimental validation. Experiments show that our model outperforms baselines, offering a solution for generating human-like spoken dialogues. Audio samples are available at https://tiamojames.github.io/DialoSpeech