There are a lot of different ways you could build it. Which way might be better for you depends on a lot of different things. Your answer is quite likely different than mine.
What are your goals? What are you trying to accomplish relative to predator protection, cost, appearance, keeping the run dry, keeping wind, snow, and ice out of the run. What are your skills. What tools and how much help will you have? How rocky is your ground? There are so many different factors involved, I'm not going to try to tell you which is best for you. I will offer a few observations though.
Your coop roof is sloping into the run. That means rain will run off into the run and could cause a problem. I'd give consideration to guttering or some other way to avoid putting water in the run, or do something to get it out of the run.
I don't know how tall you are, but I think you will greatly appreciate being able to walk in the run without banging your head. If you are going to slope it, all of it does not need to be high enough to walk under it the entire width, but the high point should be. Don't forget that when you step, your head does not stay horizontal. It bounces up and down. Give yourself some extra head height. And don't forget the the thickness of the supports for the top, if you put a top on it. The height to keep from banging your head is the bottom of the lowest thing in there, not just the height of the roofing material.
I think you are going to cover it since you sloped it. If you use a solid covering, that slope will help get the rain off instead of it setting up there on a flat roof and probably leaking.
If you do not plan on covering it, chickens will often fly up to a solid top rail just to perch. When they do that, you have no idea which side they will fly down on. No, they do not understand the concept of flying back up there to get back in. Once they do that and get out, they are effectively locked out. If you don't cover it, I suggest the top rail be about a foot or so below the top of the wire so they don't have a solid place to land.
I don't know how much space you actually have available for a run, but I always suggest giving them as much as you can. That usually makes it less work for you to do proper maintenance. How your build it will also influence this too, but a lot of building material comes in 4 and 8 foot standard lengths. With certain types of construction, you migth be able to make your run a little bigger without really spending any extra money and with virtually no extra work. I'm thinking mostly of runs that are basically wire and posts more than those with a lot of horizontal wooden segments, but it is a general concept to keep in mind.
If you are covering it, especailly with a solid roof, don't forget snow and ice load if you live were that is important. Wire can pick up ice load and then snow load too, so don't get too complacent on building it strong enough. Another important consideration along these lines. The wider you build it, the bigger the members spanning that width need to be. It is generally better to build it longer and narrow instead of short and wide if you are going to cover it.
I can't see under the overhang on you coop, but if you don't have it, I'd take advantage of that overhang and provide a lot of ventilation under there where rain won't blow in. Maybe cut out a long narrow strip and cover it with hardware cloth to keep predators out.
Good luck with it. I think it will turn out nice.