My mint has done beautifully here. It survives our winters, too. I'm in Florida. Bee Balm is also another good suggestion. And, of course, I did forget to suggest that she check her growing zone.
Don't put azaleas near your coop. I'm pretty sure it's toxic. But there are lots of vines that do well in the shade. Malabar spinach (edible by both humans and chickens) is fond of partial sun. Also, consider mint, lemon balm, and nasturtiums. - Be careful with the mint though. It has a tendency...
This too, before beautifying the fence, consider covering the fence with 1/4 inch hardware cloth and covering the run. Also, consider a predator apron. And, growing vines on the fence will help provide shade - and hide the chickens from predators - as well as look pretty
I think some vines that the chickens could eat might look nice - I'm going to plant malabar spinach on mine. It has red stems and pretty green leaves. I'm hoping mine looks this good! I actually started the seedlings yesterday.