We have studied the self-focusing and filamentation of vortex laser beams propagating in underdense plasmas with different values of the topological charge and initial laser powers. The self-focusing dynamics of vortex laser beams is closely related to the topological charge, which has attracted widespread attention. Based on the paraxial approximation of the Helmholtz equation, the steady-state solution of vortex beams propagating in underdense plasmas is deduced, and the expression for critical power of vortex laser beams is obtained. Furthermore, using the split-step Fourier method to solve the wave equation, we analyze the numerical images of vortex laser beams propagating in underdense plasma under different parameters. The simulation results show that the critical power for self-focusing is positively correlated with the topological charge of vortex laser beams. When the initial laser power is high enough, the vortex laser beam will first be focused into a thin ring, and then the modulation amplitude increases continuously, which eventually leads to the ring structure breaking into filaments. The number of filaments has an integer multiple relationship with the topological charge. In the process of filamentation, the radius and the maximum light intensity of vortex laser beam both change drastically. After the filamentation process is completed, the vortex laser beam continues to propagate with a new topological type. We further increase the incident laser power and find that the number of filaments of the vortex laser beam increases. The increased number of filaments is the value of its topological charge at each time. With the development of filament instability, higher-order modulation instability can be excited in the later stage, and the intensity of filaments will exhibit angular modulation. Our results show that in compared with the standard Gaussian beam, the propagation behavior of vortex laser beams in underdense plasmas is much more stable under the same power, wavelength and plasma parameters. The propagation characteristics of vortex laser beams are helpful to the theoretical and experimental study of stimulated backward Raman amplification of ultra-strong vortex beam in underdense plasmas.