To evaluate the continuous and instantaneous load capability of a battery, this paper describes a joint estimator for state-of-charge (SOC) and state-of-function (SOF) of lithium-ion batteries (LIB) based on Kalman filter (KF). The SOC is a widely used index for remain useful capacity left in a battery. The SOF represents the peak power capability of the battery. It can be determined by real-time SOC estimation and terminal voltage prediction, which can be derived from impedance parameters. However, the open-circuit-voltage (OCV) of LiFePO4 is highly nonlinear with SOC, which leads to the difficulties in SOC estimation. To solve these problems, this paper proposed an onboard SOC estimation method. Firstly, a simplified linearized equivalent-circuit-model is developed to simulate the dynamic characteristics of a battery, where the OCV is regarded as a linearized function of SOC. Then, the system states are estimated based on the KF. Besides, the factors that influence peak power capability are analyzed according to statistical data. Finally, the performance of the proposed methodology is demonstrated by experiments conducted on a LiFePO4 LIBs under different operating currents and temperatures. Experimental results indicate that the proposed approach is suitable for battery onboard SOC and SOF estimation.