Kayaköy, or Levissi as it was known to its former inhabitants, was a Greek town until 1923, when, after the multinational Ottoman Empire drew to close, governments of Greece and Turkey concluded on a population exchange to become nation states ethnically homogenous as much as possible on the basis of Treaty of Lausanne. According to that treaty, all Greek Orthodox inhabitants of Turkey were to be exiled to Greece, while all Muslim inhabitants of Greece were to be exiled to Turkey in return. When the Greek inhabitants of Kayaköy left for Greece, Muslims from Greek Macedonia were settled in their place. However, Macedonians who were used to large and fertile fields in their former land found this hilly and rocky area with little arable land unfit to live, and abandoned the place in favour of other regions, and thus Kayaköy has become what it is today.
In its heyday, Kayaköy was populated enough to support a local newspaper and several schools and stores, but today there is only a handful of natives living there, mostly in the neighbourhood of Keçiler, 2 km north of the "ghost town" of Kayaköy.
You'll mostly walk around the ghost town.
The ghost town of Kayaköy itself, including hundreds of abandoned houses with no roofs or windows, and the walls of some of which are partially ruined, is the main sight. Those not to be missed include the old fountain which dates back to 1888 by the tarmac road, two abandoned churches (aptly dubbed Yukarı Kilise and Aşağı Kilise, i.e. "Upper" and "Lower Church", respectively, because of their relative elevation difference to each other), and the little chapel on the top of the hill (about 20 min uphill walk from the lower church; follow the red dots from the church), which gives a stunning view of the valley and the sea below, which are located on the other side of the hill that Kayaköy leans against, and therefore is not visible from the town itself.
The lower church and the streets (or stairs to be more precisely) in its vicinity has been declared a "museum" by the Turkish Ministry of Culture with a ticket office in the entrance which require anyone passing through—whether they have the intention to check out the church or not—to buy a ticket which costs 8 TL pp. You may pass on the early and late hours of the day on which the office is unmanned without buying a ticket, though.
Hiking to Cold Water Bay
Not just for hiking but also for a dip in the sea, you can take the marked trail to Cold Water Bay (Soğuksu Koyu), just south of ghost town, but not immediately visible because of the hills Kayaköy leans against. While it is not much more than a cove with a quite short of stretch of pebble beach really, it is the nearest access to the sea from Kayaköy. The restaurant with wild boar stew on the menu just behind the beach may also worth the trip there.
The waymarks on this route are red dots. There are no signs but the dots are placed once every few metres, so it's upsettingly impossible to get lost.
The route starts from in front of the lower church and ascents towards the hilltop. To avoid the fee of 8 TL to access the area around the church, either start very early before the office opens, or take a long detour around the part of the town that was declared museum (you can start by taking the path on the side of the old fountain). Take the small chapel on the very top hill as a bearing — you should be walking towards (but not straight to) it. Once you are clear off the house ruins, near the hilltop, there are two trails: One leading to the chapel, other descending towards the (now visible) sea. Both trails are signed with the same red dots so be careful. At this point, the trail begins descending on volcanic-looking somewhat slippery surface between again volcanic-looking huge brown rocks. After passing a short level ground with some shrubs here and there, it again starts descending towards the sea, which the trail soon meets.
There are restaurants in and around the town, mainly between Keçiler and Kayaköy.
You'll find a bar or two situated in the yards of the abandoned houses close to the church.
There is a number of guesthouses (pansiyon) both in and out of the ghost town. There are also some bungalows which start from 20 TL a night.
The town is within the GSM coverage, no matter if it is a ghost town or not.
Two marked hiking trails head out from Kayaköy (apart from the one that descends to Cold Water Bay) — one of them descends from the upper church towards the shore and Ölüdeniz, while the other one leads to Ovacık inland, north of Ölüdeniz, the official trailhead of Lycian Way.