The initial and boundary condition, the hydrodynamic force on the cylinder surface and the cylinder response equations with shear flow were derived based on the stream function-vorticity equations in the exponential-polar coordinates attached on the moving cylinder. The whole evolutions of cylinder starting from rest and then undergoing development and vibration steady were calculated and discussed. The development process of separation vortexes, the deformation and shift of drag-lift phase diagram and the variation of cylinder vibration and equilibrium position were described. Moreover, the steady condition of vortex-induced vibration with the shear rate K was investigated. The results show that the symmetrical flow field will be broken due to the background vorticity generated by the shear flow which also causes the increase of upper vortex strength and the decrease of lower vortex strength. The vortex street inclines toward the lower side and the inclination of vortex streets increase with the increasing shear rate K. So does the shift of pressure curves which leads to the increase of absolute value of lift, the amplitude and the shift of cylinder.