Drone-mounted base stations (DBSs) are promising solutions to provide ubiquitous connections to users and support many applications in the fifth generation of mobile networks while full duplex communications has the potential to improve the spectrum efficiency. In this paper, we have investigated the backhaul-aware uplink communications in a full-duplex DBS-aided HetNet (BUD) problem with the objective to maximize the total throughput of the network, and this problem is decomposed into two sub-problems: the DBS Placement problem (including the vertical dimension and horizontal dimensions) and the joint UE association, power and bandwidth assignment (Joint-UPB) problem. Since the BUD problem is NP-hard, we propose approximation algorithms to solve the sub-problems and another, named the AA-BUD algorithm, to solve the BUD problem with guaranteed performance. The performance of the AA-BUD algorithm has been demonstrated via extensive simulations, and it is superior to two benchmark algorithms with up to 45.8% throughput improvement.