This study investigates wireless information and energy transfer for dual-hop amplify-and-forward full-duplex relaying systems. By forming energy efficiency (EE) maximization problem into a concave fractional program of transmission power, three relay control schemes are separately designed to enable energy harvesting and full-duplex information relaying. With Rician fading modeled residual self-interference channel, analytical expressions of outage probability and ergodic capacity are presented for the maximum relay, signal-to-interference-plus-noise-ratio (SINR) relay, and target relay. It has shown that EE maximization problem of the maximum relay is concave for time switching factor, so that bisection method has been applied to obtain the optimized value. By incorporating instantaneous channel information, the SINR relay with collateral time switching factor achieves an improved EE over the maximum relay in delay-limited and delay-tolerant transmissions. Without requiring channel information for the second-hop, the target relay ensures a competitive performance for outage probability, ergodic capacity, and EE. Comparing to the direct source-destination transmission, numerical results show that the proposed relaying scheme is beneficial in achieving a comparable EE for low-rate delay-limited transmission.