We study the dynamics of single electron in an inhomogeneous cylindrical plasma channel during the direct acceleration by linearly polarized chirped laser pulse. By adjusting the parameters of the chirped laser pulse and the plasma channel, we obtain the energy gain, trajectory, dephasing rate and unstable threshold of electron oscillation in the channel. The influences of the chirped factor and inhomogeneous plasma density distribution on the electron dynamics are discussed in depth. We find that the nonlinearly chirped laser pulse and the inhomogeneous plasma channel have strong coupled influence on the electron dynamics. The electron energy gain can be enhanced, the instability threshold of the electron oscillation can be lowered, and the acceleration length can be shortened by chirped laser, while the inhomogeneity of the plasma channel can reduce the amplitude of the chirped laser.