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.
Afkule and Gemile
Southwest of Kayaköy, the beach at Gemile is accessible by a gravel road good enough for most conventional cars from the neighbourhood of Kınalı, about 2 km west of the ghost town, although there is no public transport heading there. There is also a hiking path which somewhat shortcuts the gravel road. The trail begins out of Kınalı, at where the road starts windings and descents towards the coast. Just off shore of Gemile is the St Nicholas Island (Gemiler Adası in Turkish) with some ruins of a Byzantine chapel which dates back to 5th century.
Perhaps a more rewarding sight in the same direction is the monastery at Afkule (also spelled Af Kule), clinging at the high cliffs over the sea, and which affords really impressive views over the Gulf of Fethiye, as far away as Rhodes if the air is clear. Other than its roof, this Greek Orthodox monastery is as sound as it was when abandoned in 1920s. To get there, you will need a short (about 3 km) but pretty demanding hike up and down along a trail, part of it, though, fortunately, through a pine forest. In total it takes around an hour on foot to get there from the ghost town. West of Kayaköy, leave the gravel road to Gemile at where the trail (a dirt track) to Afkule branches off, which is properly marked by yellow signs. At about midway through the trail to Afkule (about 1 km away from where you started), you'll notice a branching track to right—this is the wrong path, keep to the trail to left instead. The trail ends in a open space—carpark for those taking their vehicles along the track. From here, take the wide track to right, which is waymarked with the yellow&red marks usual in the area, and which will slowly ascent to the top of the cliff, which is about 1 km away from the carpark. Then the trail will descend down to the monastery, but be extremely careful in this section as it is very easy to slip down since the path is covered with loose gravel. Perhaps not for the faint hearted, you can further explore the lower stories of the complex (the church and residences of the monks) once you are in the monastery, though that will require a very steep climb down with the full view of high cliffs down to the coast (a whopping 400 mt just below your feet), so those with afraid of heights—do beware.
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.