In this paper, a multiscale physics-informed neural network (MPINN) approach is proposed based on the regular physics-informed neural network (PINN) for solving stiff chemical kinetic problems with governing equations of stiff ordinary differential equations (ODEs). In MPINNs, chemical species with different time scales are grouped and trained by multiple corresponding neural networks with the same structure. The adaptive weight based on a key performance indicator is assigned to each loss term when calculating the summation of loss residues. With this structure, MPINNs provide a framework to solve challenging stiff chemical kinetic problems without any stiffness-removal artifacts before training. In addition, by introducing a small number of ground truth data (GTD) points (less than 10% of the number required for residual loss calculation) and adding data loss terms into loss functions, MPINNs show superior ability to represent stiff ODE solutions at any desired time. The accuracy of MPINNs is tested with classical chemical kinetic problems, by comparing with the regular PINN and other state-of-the-art methods with special consideration for solving stiff chemical kinetic problems with PINNs. The validation results show that MPINNs can effectively avoid the influence of stiffness on neural network optimization. Compared with the traditional deep neural network only trained by GTD, MPINNs can use no data or a relatively small amount of data to achieve high-precision prediction of stiff chemical ODEs. The proposed approach is very promising for solving stiff chemical kinetics, opening up possibilities of MPINN application in different fields involving stiff chemical dynamics.