[Growler]

Aerate and turn over the soil in the grassy area you want to keep. Dog droppings should be removed, not left to rot - unless you want to turn them into the soil right away. Excavate the patio area to a depth of at least 8" and add a weed-barrier membrane to the very bottom of the patio pit - porous to water filling it up, but a barrier to weeds growing in the cracks from below. Level as best you can, with a minor grade away from the house - it won't do to have water from ice/snowmelt and/or rain pooling right against your home's foundation.

I'd leave some grass if you ever want an easy time selling the house someday.