I like Grandpa’s idea because it’s close to what I did. Well maybe not that close. Install four posts and level them. Make a rectangular frame to set the coop on. Fasten it all together so the wind doesn’t blow it off. There are several different ways to do that. Depending on the size and slope, TwoCrows idea works too. There are always different ways to solve these things. How rocky the ground is and how steep that slope is are factors.
To support a 4’ x 8’ Grow-out coop maybe 18” off the ground, I drove 4' long sharpened 2x4’s into the ground with a sledge hammer and got them level that way, then built a 2x4 frame on top of those to set the coop on.
I did not use any diagonal bracing but if your two downside posts are very high, it’s a great idea. I’m a structural engineer and triangles really stiffen things up.
The main reason I’m posting is that I think you will have another problem if you build a run. Erosion. Chickens will clean off any vegetation that is there and the soil will erode away in a heavy rain. Their scratching will move that dirt downhill too.
Someone had some nice photos a while back on this. They solved the problem by installing timbers across the slope from one side of the run to the other to sort of form terraces. Their run was basically a series of steps. A timber across with level dirt behind it, then another timber with level dirt behind that.