This paper develops a novel approach for online Lithium-ion (Li-ion) battery model identification and state of charge (SOC) estimation. To account for the SOC-dependent battery dynamics and the static nonlinearity between the open-circuit voltage (OCV) and SOC, we formulate a grey box nonlinear state-space model, in which elements depend on SOC in a polynomial way. For model identification, we propose an online concurrent state and parameter estimation by alternating the recursive least squares algorithm and particle filter; the SOC is computed via Coulomb counting during the modeling. The identified grey box model is then applied for SOC estimation using the particle filter. Simulation with real-world battery measurements demonstrates the effectiveness of the model structure and the estimation approach, which is reflected in accurate terminal voltage estimation and nonlinear OCV-SOC relation, and superior performance regarding SOC estimation compared to state-of-the-art approaches.