We consider a dual-hop full-duplex relaying system, where the energy constrained relay node is powered by radio frequency signals from the source using the time-switching architecture, both the amplify-and-forward and decode-and-forward relaying protocols are studied. Specifically, we provide an analytical characterization of the achievable throughput of three different communication modes, namely, instantaneous transmission, delay-constrained transmission, and delay tolerant transmission. In addition, the optimal time split is studied for different transmission modes. Our results reveal that, when the time split is optimized, the full-duplex relaying could substantially boost the system throughput compared to the conventional half-duplex relaying architecture for all three transmission modes. In addition, it is shown that the instantaneous transmission mode attains the highest throughput. However, compared to the delay-constrained transmission mode, the throughput gap is rather small. Unlike the instantaneous time split optimization which requires instantaneous channel state information, the optimal time split in the delay-constrained transmission mode depends only on the statistics of the channel, hence, is suitable for practical implementations.