If you re do the patio area into an at grade deck, one step down from house, and up from the backyard grade, you could mirror the aspect ratio design dimensions of the house, and make it longer and narrower to suit the space. Then, the outside area could be done in an artificial turf product such as K-9 grass, or Ultimate Grass, which doesn't require sun! Also, if you take a corner to designate it the area in which you want to dogs to do their business, you could get a large piece of driftwood and delineate the space using a different flooring... such as sand or even mulch. Dogs will typically mark the driftwood, and they could be trained to use this area almost exclusively. Adding a decorative screen in front of that space, or a small tree, would help to create some vertical interest in the back yard, and places for birds to roost and nest.

jordanmederos, adding artificial grass will be a solution for a backyard like yours that gets a lot of shade, just make sure that it's designed for pets. The dogs won't destroy it, and it will add some color to your space. Do your research on products since not all are created the same. Good luck with your new backyard, and post some pics of how it turns out!