Microchannel flow with electrolyte solution is often influenced by the presence of a double layer of electrical charges at the interface between the liquid and the wall of a substrate. These surface interactions affect strongly the physical and chemical properties of fluid and substantially influence the heat, mass and momentum transport in microfluidic systems. Traditional computational fluid dynamics methods using the modified Navier-Stokes equation for electrokinetics in solving macroscopic hydrodynamic equations have many difficulties in this area. We present here a lattice Boltzmann model in the presence of external force fields to describe electrokinetic microfluidic phenomena using a Poisson-Boltzmann equation. Pressure is considered as the only external force to drive liquid flow in microchannels. Our results from a 9-bit square lattice Boltzmann model are in excellent agreement with recent experimental data in pressure-driven microchannel flow that could not be fully described by electrokinetic theory. The differences between the predicted and experimental Reynolds numbers from pressure gradients are well within 5%. Our results suggest that the lattice Boltzmann model described here is an effective computational tool to predict the more complex microfluidic systems that might be problematic using conventional methods.