First, you want a grass that is tough and suited for that area. Assuming it gets sun, my choice would be a good turf type tall fescue. Normally, fescue is a clump type grass, meaning the plant grows in one clump and won't send out rhizomes to spread. However, there are now some varieties available that have been developed to spread. They still don't spread as readily as bluegrass, but they will spread to fill in bare spots over time, which a regular fescue won't. RTF is one variety, you can search for it on the web. I've also seen it at HD. That could be a good choice. Of course as others have pointed out, if the dogs are on it a lot, there maybe areas that you really can't maintain.
If you go that route, I would probably wait till fall, which is optimum time, then kill what's there, and use a core aerator, then a slice seeder to establish the new lawn.