How tall is the hardware cloth you plan to use? How tall is the run? If the run is tall enough for you to stand in there, and I highly recommend that, you will probably be overlapping the hardware cloth. There are different ways to do that of course. You can use J-clips, hog rings, or wire. I've used all of them. My preferred way when I can is to sandwich the edges between two pieces of wood screwed together. If you clamp it between two pieces of wood with screws going through the holes in the hardware cloth it will not go anywhere. Drill pilot holes so you don't split the wood.
Are you planning on stretching it? My stretcher is two 2x4's clamped over the end with bolts, washers, and butterfly nuts. I use a flexible cable attached to both ends I can put a sliding hook through. That way it self-adjusts to get equal tension of both top and bottom of the fence. Stretching it even a little can get a lot of the flex out of it. Just stretch it in a straight line, do not try to stretch around a corner post or you will break that post off. Don't ask how I know.
Where is the run door going to be, coming from the coop or will you be able to enter the run from outside the coop? That might make a difference on your post spacing.
As Wyodreamer sort of said, you can go vertical with the wire or horizontal. Are you planning on some type of solid roof or a wire roof? To darn many options on how you could be building it. If I were building what I think you might be with the wire going horizontal, I'd set posts at 8' since the least expensive lumber tends to come in 8' lengths. I think you will be running lumber horizontally between posts. I'd sandwich the ends where they meet depending on overall height and width of your sections between wood and be comfortable with an 8' horizontal run between posts. The shorter span the other way will give you a lot of stiffness.
Some people use 1/4", 1/2" or even 1" hardware cloth. Snakes big enough to eat eggs or baby chicks can get through the 1". I'm happy with 1/2" but a few things like mice can get through that. This is the type of thing that depends on your opinion, and those can vary.
