This paper investigates the cross-layer optimization of fast video delivery and caching for minimization of the overall video delivery time in a two-hop relaying network. The half-duplex relay nodes are equipped with both a cache and a buffer which facilitate joint scheduling of fetching and delivery to exploit the channel diversity for improving the overall delivery performance. The fast delivery control is formulated as a two-stage functional non-convex optimization problem. By exploiting the underlying convex and quasi-convex structures, the problem can be solved exactly and efficiently by the developed algorithm. Simulation results show that significant caching and buffering gains can be achieved with the proposed framework, which translates into a reduction of the overall video delivery time. Besides, a trade-off between caching and buffering gains is unveiled.