It looks like you had an original deck and someone added another section. IMO it will be a waste of money to do a patio under there with all the posts under there unless you reconfigure framing and remove some of the posts if you want to keep the original footprint.

Please show a picture from the back of the house so you can better understand what the back of the house looks like, maybe adding a section across the back and re-doing the existing one it will give you a much more functional space up and down.

Not to mention all the debris and water from the deck will make the patio virtually useless and spending the dollars on a ceiling system for that deck would not be worth it is my opinion.