The conformational and rheological dynamics of dilute solutions of polyelectrolyte macromolecules undergoing shear and extensional flow are modeled using Nonequilibrium Brownian Dynamics (NEBD) and Configuration-Biased Monte Carlo (CBMC) simulations. The mathematical model utilizes a bead-spring chain with charged beads that interact through a screened Debye-Huckel potential, and that also interact through stretching and bending forces. The diffusion (or Fokker-Planck) equation for the probability density of the positions of the beads of the chain is converted to a Stochastic Differential Equation (SDE), from which the simulation algorithm for the NEBD is obtained. The CBMC is used in the initial chain generation and in determining steady-state properties in elongational flows. Various conformational and theological quantities, such as the stress and birefringence, are monitored, under both steady-state and transient conditions, with the primary independent variable being the salt concentration (parametrized through the Debye length) and the strength of interact:ion q, related to the degree of ionization of the chain. It is found that this model is able to describe qualitatively many of the experimentally observed features in such systems.