The cold-start recommendation is an urgent problem in contemporary online applications. It aims to provide users whose behaviors are literally sparse with as accurate recommendations as possible. Many data-driven algorithms, such as the widely used matrix factorization, underperform because of data sparseness. This work adopts the idea of meta-learning to solve the user's cold-start recommendation problem. We propose a meta-learning based cold-start sequential recommendation framework called metaCSR, including three main components: Diffusion Representer for learning better user/item embedding through information diffusion on the interaction graph; Sequential Recommender for capturing temporal dependencies of behavior sequences; Meta Learner for extracting and propagating transferable knowledge of prior users and learning a good initialization for new users. metaCSR holds the ability to learn the common patterns from regular users' behaviors and optimize the initialization so that the model can quickly adapt to new users after one or a few gradient updates to achieve optimal performance. The extensive quantitative experiments on three widely-used datasets show the remarkable performance of metaCSR in dealing with user cold-start problem. Meanwhile, a series of qualitative analysis demonstrates that the proposed metaCSR has good generalization.