Hot-wire directed energy deposition using a laser beam (DED-LB/w) is a method of metal additive manufacturing (AM) that has benefits of high material utilization and deposition rate, but parts manufactured by DED-LB/w suffer from a substantial heat input and undesired surface finish. Hence, regulating the process parameters and monitoring the process signatures to control the final quality during the deposition is crucial to ensure the quality of the final part. This paper explores the dynamic modeling of the DED-LB/w process and introduces a parameter-signature-quality modeling and control approach to enhance the quality of modeling and control of part qualities that cannot be measured in situ. The study investigates different process parameters that influence the melt pool width (signature) and bead width (quality) in single and multi-layer beads. The proposed modeling approach utilizes a parameter-signature model as F1 and a signature-quality model as F2. Linear and nonlinear modeling approaches are compared to describe a dynamic relationship between process parameters and a process signature, the melt pool width (F1). A fully connected artificial neural network is employed to model and predict the final part quality, i.e., bead width, based on melt pool signatures (F2). Finally, the effectiveness and usefulness of the proposed parameter-signature-quality modeling is tested and verified by integrating the parameter-signature (F1) and signature-quality (F2) models in the closed-loop control of the width of the part. Compared with the control loop with only F1, the proposed method shows clear advantages and bears potential to be applied to control other part qualities that cannot be directly measured or monitored in situ.