This work proposes a methodology for the energy-and cost-efficient 3-D deployment of an unmanned aerial vehicle (UAV)-based aerial access point (AAP), that exchanges a given amount of independent data with a set of ground user equipment (UE). Considering a fly-hover-communicate transmission scheme, the most energy-efficient 3-D hovering points (HPs) of the AAP are determined by decoupling the problem in the horizontal and vertical dimensions. First, we derive analytically the optimal hovering altitude that jointly maximizes the downlink and uplink global energy efficiency (GEE) of the system. Next, we propose the multilevel circle packing (MCP) algorithm to determine the minimal number of HPs and their associated horizontal coordinates, such that the AAP covers all the UEs in the given geographical area. A cost analysis is carried out to observe the variation of both fixed and variable costs; these are then minimized by suitably selecting the AAP's battery parameters, like the depth of discharge (DOD), defined as the portion of battery capacity that is consumed during a discharge cycle, and the velocity of the UAV. Simulation results show that: the UAV energy consumption has a significant impact on the 3-D HPs of the AAP; the time spent during the substitution swap of an out of power AAP has a major influence on the operational cost; the cost of the system can be optimized by suitably selecting the onboard battery and the UAV flight parameters.