Direct air-to-ground (A2G) communications leveraging the fifth-generation (5G) new radio (NR) can provide high-speed broadband in-flight connectivity to aircraft in the sky. A2G network deployment entails optimizing various design parameters such as inter-site distances, number of sectors per site, and the up-tilt angles of sector antennas. The system-level design guidelines in the existing work on A2G network are rather limited. In this paper, a novel deep learning-based framework is proposed for efficient design and optimization of a 5G A2G network. The devised architecture comprises two deep neural networks (DNNs): the first DNN is used for approximating the 5G A2G network behavior in terms of user throughput, and the second DNN is developed as a function optimizer to find the throughput-optimal deployment parameters including antenna up-tilt angles and inter-site distances. Simulation results are provided to validate the proposed model and reveal system-level design insights.