When the chaser spacecraft closely tracks and observes the non-cooperative target in space, it should approach to and keep pointing to the non-cooperative target from the particular direction. For the problem that the chaser spacecraft keeps closely tracking and pointing to the non-cooperative target, in the case of the target with the attitude motion and the unknown orbit maneuver, based on the relative orbit dynamics and the attitude dynamics which are described in the line-of-sight coordinate frame and the body coordinate frame, respectively, the six-degree-of-freedom model of orbit and attitude simultaneously control is proposed. The RBF neural network is employed to adaptively estimate and compensate the system uncertainties and the unknown motion parameters of the target. Using the backstepping method, a controller which can control the chaser spacecraft to converge to the desired relative orbit and attitude in finite time is proposed. Considering the nonlinearity of the control input, such as saturation and dead zone, an improved control algorithm is developed. The simulation results are provided to show the effectiveness of the control model and the control algorithms. Moreover, the improved control method has higher control accuracy.