How should I clean and repaint a painted wooden deck?
Painting a pressure treated wood deck is always tricky. Pressure treated wood does not seem to hold paint well, even with a good primer. Stain is usually a better alternative, but since your deck already has paint, it would have to be stripped completely before using stain. 

In order for the paint to stick fairly well, the surface needs to be as clean and dry as possible. Looking at your pictures, I see a fair amount of algae in the grain of the wood. This is a very normal condition. You should however, use the pressure washer and bleach/water/TSP shrub down method to clean it up. I'd wait until April or May when temps are up in the 60's to 70's to help dry the deck and open the grain. 

Select a high grip primer rather than a stain blocking type. Many manufactures have come out with new formulas that are designed to bond better. When you apply the primer, don't rush. Simply rolling on primer too quickly is not good. The primer needs to be brushed into the grain as deeply as possible.  Let it dry completely.  Now you can apply your deck paint as normal.