First, brush very dirty windows using a soft brush. As such, the coarse dirt will not simply be spread all across the pane when it is subsequently soaped. Water with a little dishwashing detergent will suffice, instead of using expensive window cleaning fluids. The water can then be stripped off using vertical strokes.
Ideal results are achieved when working in slightly sunny weather. The soft lighting makes streaks visible and thus helps when giving the windows a polishing finish. Strong sunshine, on the other hand, dries up the detergent too quickly, thus leaving behind streaks on the pane.
In order to avoid streaks when cleaning windows, use newspaper to give the panes a polishing finish and remove the cleaning water.