The integration of simultaneous wireless information and power transfer (SWIPT) and mobile-edge computing (MEC) technologies is emerging as a promising technique to overcome the performance limits of ultra-low power devices (ULPD) due to their low battery capacities and their limited computation capabilities in the Internet of Things (IoT) era. In this paper, we propose an online resource allocation algorithm for multiuser SWIPT-based MEC systems with the aim of maximizing the proportional fairness computational utility function subject to the stability of task and energy queues. Lyapunov optimization framework is used to jointly optimize the amounts of time allocated for energy harvesting, information decoding and offloading, the transmission power for offloading and CPU-cycles frequencies for local computing. Moreover, rigorous performance analysis has been done to prove the asymptotic optimality of our proposed algorithm. Simulation results are also presented to demonstrate the gains of our proposed algorithms over alternative online approaches and the impact of different network parameters on the performance of our algorithm.