There are plenty of shade plants (annuals and perennials, even climbing plants to use your trellis) that you could grow there, and along with mulch, should reduce the weed problem.

However, if you don't go for that route, maybe you can also use a weedeater to take care of the weeds and plants that the mower can't get to.

How wet is the area? You could plant ferns. You could plant a vine (eg Trumpet Vine) that could climb over the deck to get sunlight, but that can be risky. (Vines can be invasive and bad for houses) Maybe have climbing roses that reach the soil there and climb a trellis above the deck?

Thanks for your suggestions. Gerdenper, your idea of using climbers on the trellis is a good one, but the height is only about 3 feet and they may become entwined in the floor of the deck and be a nuisance. The area is north facing and gets very little sun, so I think Ed's suggestion of ferns would be my best bet. I do have some ferns at the back of my shade garden which I could split and once I get them in, they should fill the bill quite nicely. Thanks for your responses. Happy summer everyone!!