How do I make a raised garden-type dog patch on my back patio ?

Hi. Congrats on the move. You will need to have the topography mapped out so you don't cause a drainage problem to your home or to your neighbors. After that's done, if water run off won't be a problem where you want to put it, build it like a regular raised garden. I'd also put in a sprinkler system that would run at night to wash urine out. Easier still would be to just take nice long walks (and do everybody good in the long run!).

Congratulations! Sounds like a good plan. I too, would be concerned about damage to the concrete. And possible staining from runoff. My first thought was to add a good layer of rocks at the bottom to help with air flow (so it does not smell) and aid in drainage. Then I read this:

https://www.houzz.com/discussions/2152057/putting-raised-beds-on-concrete-greenhouse-on-concrete

Which confirmed my rock theory.

Here is another site:

https://learn.eartheasy.com/articles/gardening-on-concrete-with-raised-beds-and-patio-containers/

At least something for you to mull over before making a decision.

Congrats! To both of you for your new nest! I live in MN with 5 dogs and a similar situation. We had to remove some of the concrete because of the grading & not wanting to have issues with that or smell. We rented a jackhammer and it went fast. We were lucky to have a community garden near by and they came & got the big concrete pieces. Then we started like we were doing a patio adding a sprinkler system to that area. Instead of pavers we went with a turf the good stuff they use on foot ball field. It was expensive but totally worth it! The poo picks up easy, the urine washes down. Twice a year I use an enzyme cleaner just cuz I wash everything back there. Amazing it’s been 9 years when we had 8 dogs and we have no wear or tear to the turf. It was a little expensive for the project but well worth it🐶