Chickens always scratch the ground while foraging for seeds, which damages the grass. If you mow the grass too short, it will be very vulnerable and the chickens may finish it off. However, if you let the clippings lay on the mowed grass, the chickens will probably prefer them over the intact plants. Source:
3 Ways to Save Your Grass When Raising Chickens
They also like to eat greens and prefer some weeds over everything else, but will not touch others. The consistency of your lawn may change due to that.
Chicken manure is one of the best natural fertilizers, but it's too rich in nitrogen for grass and causes chemical burns on the plants and roots. You should divide your yard into several areas and let the chickens graze in one part at a time, then move them into the next part each day. This gives the grass some fertilizer (but not too much) and spreads the manure evenly.
Chickens will deliberately scratch patches of land bare because they want a nice sand bath to keep clean of parasites. Providing them with a big bowl of loose and dry sand can prevent excessive digging. Source: Will chickens kill my grass and ruin my lawn?
As said in the comments, a chicken never wants to be alone. You need at least 2 of them to cater to their social needs.
You do realize that chickens are birds, right? Chickens can fly, even if they're not the best at it. Some breeds can easily fly over 3 - 4 feet fences. Your ex-battery hen might not know how to fly, but she might also discover it one day. You'll either need to cover your backyard with netting or keep them in a coop like this one. The advantage of such a coop is that you can move it every day to spread the manure and give the grass a rest. The other alternative is to cut their flying feathers short, which needs some training to be done safely and must be repeated after each molt.
You'll also need a hen hut where they are safe at night. There are some portable chicken tractors (another example) that combine coop and house and are easily movable on wheels, but they might be expensive. On the other hand, you'll need some sturdy fencing if there are foxes, coyotes, possums, or raccoons around that might want to get a snack from your hen house.
If your coop isn't connected to the hen house, you'll want to give your hens a little hideout. Just 2 wooden boards screwed together at an angle and big enough for your hens to hide underneath is enough. It's their instinct to hide under bushes and branches whenever they see a bird of prey circling the sky. Without such a hideout they might panic.
And lastly, don't forget a chicken water feeder. Those aren't expensive and you can even craft your own, but it might take an ex-battery hen some getting used to it.