A Hawkman and Hawkgirl series where the characters share the marquee, to begin with. These are two characters whose relationship DC should be highlighting right now, not discarding. I've said it before and I'll say it again: Marvel have Reed and Sue, DC should have Katar and Shayera as their premiere, rock-solid power couple. I guess Aquaman and Mera sort of hold that title now, but why not one-up 'em, DC, and just have two?
To be more specific, though, considering the current state of the Hawks, I'd take Hawkman back to Thanagar and begin to rebuild the character there, not so much from the ground up, but rather by reinforcing the frame that's already been erected. He's been through a lot since The New 52 first started, but no one ever really bothered to decide just who he is exactly. And when you're lost, sometimes the best way to find yourself is to go back to the beginning. So I'd take Katar back home to Thanagar and, through a series of flashbacks and self-reflection, establish his new origin in a more definitive way. Why was he even on Earth to begin with, and what happened to his memories? These are things that were never competently explained, and I know just how to do it. His history as a gladiator, which was touched upon briefly in The Savage Hawkman, is something worth exploring, too, as well as him being an adopted son of royalty. Both of those aspects have potential, I think; they just need to be fleshed out a lot more. And not only would this help to build up Hawkman, but also Thanagar, which we know very little about so far.
Central to all of this, of course, would be Shayera. I'd reincorporate her right away, in fact, even using her as the catapult for Katar's return home and self-reflection in the first place. Her "death" was ambiguous at best, so I wouldn't even make a big deal over bringing her back. Just say she was found alive floating in deep space in stasis and render her as a redhead, no questions asked.
I'd conclude the first arc, then, with her coming around, completing Katar's journey of self-discovery and purpose. And that purpose? Returning Thanagar to a more stable planet. He and Shayera both are royalty now. I'd play that up. In the same way that Aquaman is the King of Atlantis, Hawkman and Hawkgirl should be the rulers of Thanagar. Basically, I'd attempt to recreate the dynamic the Silver Age Hawks shared, but in a new way. They aren't just good cop and bad cop now: they're benevolent leader and would-be tyrant. Hawkman's peace-through-strength attitude would constantly be challenged by Shayera's more measured approach to ruling, and it would be that compromise that would ultimately make them good, if not very flawed leaders trying to reconnect with their planet and its people.
Finally, I'd hope to keep them secluded on Thanagar for a good long while, separate from the rest of the superhero shenanigans going on in the rest of the DC Universe. Eventually, I'd like to take them to Earth and see them placed in a team setting; I think that's crucial for the characters' long-term success. At the outset, though, I'd prefer to be left to my own devices so as to build up their backstories and supporting cast, as well as the Thanagarian geography and culture. The planet itself, by the way, should not be neglected. Thanagar is an entire fictitious planet, after all. Think about all of the possibilities there are to explore, and as the reigning rulers of it all, there's a perfect excuse for Katar and Shayera to take a globetrotting tour of it, restoring peace to their kingdom one problem at a time. And if editorial wanted to tie the Hawks to the greater DC Universe in the interim, I'd capitalize on some of their more cosmic-oriented characters and groups, such as the Green Lantern Corps, Lobo, and maybe even L.E.G.I.O.N. or R.E.B.E.L.S., either of which I'd love to see resurface soon.
My primary goal, though, would be to make the book about Katar and Shayera traveling from place to place, partaking in varied adventures as husband and wife. The quiet moments between them would be as central to their stories as the explosive fights they often found themselves in. This is what I really enjoy about the old Silver Age Hawkman and Hawkgirl stories, and it's what I believe makes them unique to the DC Universe. Even in Hawkman Volume 4, I think you'll find that the best issues are where Hawkman and Hawkgirl are out and about, interacting with some strange new set of circumstances as two parts of a greater whole. It's their niche, and it should be at the forefront of any book they star in, regardless of anything else.