Dilute polyelectrolyte solutions show viscoelastic behaviour different from solutions of uncharged polymers by virtue of the additional stiffness provided by the repulsion between charges on the polymer chain and screening of the charged sites by ionic species in the system. Experiments report an increase in chain extensions and polymeric viscosities due to the charged nature of the polymers. These effects impact the macroscopic flow behaviour of polyelectrolytes in complex geometries. In this work, we have developed a theoretical model describing the viscoelastic behavior of dilute polyelectrolyte (PE) solutions using Onsager’s variational principle. The PE molecules are described as finitely extensible charged dumbbells suspended in a solvent containing ionic species which screen the charged sites on the macromolecules. We first formulate the rheological model in a conformation tensor formalism and couple it to the steady Stokes equations to simulate the flow of dilute polyelectrolyte solutions in confined geometries. Our results indicate an increase in the drag experienced by the bodies in a flowing PE solution, increased pressure drops and normal stresses with increasing molecular charge density. Next, the model is reformulated to consider stress induced inhomogeneities in the system and the effect of counterion fluctuations on molecular migration of polyelectrolytes is investigated. We conclude that our model based on a simple charged dumbbell with counterion screening, together with Onsager’s variational principle is a powerful tool to investigate the dynamics of dilute polyelectrolyte solutions and the effects of the polyelectrolyte charge and solvent salinity.