This paper presents the first mathematical formulation for the effect of double layer deformation on the conformational and rheological behavior of a dilute solution of flexible polyelectrolytes. The polyelectrolyte is modeled by a charged FENE dumbbell, in which the bead centers are regarded as charge points, each carrying an equal effective charge, in the calculation of electrostatic interaction under the Debye-Huckel approximation. For an arbitrary linear flow, an induced electrical force on each bead associated with the double layer deformation is derived. Incorporating this force in the diffusion equation and using a perturbation method, the configuration distribution function of the dumbbells can be determined analytically for the case of an extensional flow. The resulting distribution function is then used to calculate the root-mean-square end-to-end distance. It is found that the double layer deformation leads to an increase in the average dumbbell length, compared to the results by ignoring this effect. The force associated with the charge cloud distortion increases with decreasing double layer thickness, as opposed to the equilibrium electrical force. The stress behavior is also discussed for the cases of thin and thick double layers.