Since reconfigurable intelligent surface (RIS) is considered to be a passive reflector for rate performance enhancement, a RIS-aided amplify-and-forward (AF) relay network is presented. By jointly optimizing the beamforming matrix at AF relay and the phase shifts matrices at RIS, two schemes are put forward to address a maximizing signal-to-noise ratio (SNR) problem. Firstly, aiming at achieving a high rate, a high-performance alternating optimization (AO) method based on Charnes-Cooper transformation and semidefinite programming (CCT-SDP) is proposed, where the optimization problem is decomposed to three subproblems solved by CCT-SDP and rank-one solutions can be recovered by Gaussian randomization. While the optimization variables in CCT-SDP method are matrices, which leads to extremely high complexity. In order to reduce the complexity, a low-complexity AO scheme based on Dinkelbachs transformation and successive convex approximation (DT-SCA) is put forward, where matrices variables are transformed to vector variables and three decoupled subproblems are solved by DT-SCA. Simulation results verify that compared to two benchmarks (i.e. a RIS-assisted AF relay network with random phase and a AF relay network without RIS), the proposed CCT-SDP and DT-SCA schemes can harvest better rate performance. Furthermore, it is revealed that the rate of the low-complexity DT-SCA method is close to that of CCT-SDP method.