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From the small town square with traffic lights (on the side of which the bus station -otogar- is located), main road ascents somewhat steeply towards the hills. About 150-200 metres away from the square, it turns left. About a further 150-200 metres away, there is a gate-like structure over the highway, proclaiming that you are now out of Kaş. This place may look like a good spot to hitch, but there is a much better one lying only 100 mt away, which is not immediately visible because of the smooth slope behind the gate. After the gate, the shoulder becomes much wider and the place is much windier as it's all open to the sea breeze - which is somewhat of a gift, because Kaş lies on what is one of the southernmost points of Turkey and the heat and humidity during summer will test your patience a little bit, especially if you are carrying a large backpack. It shouldn't take more than 20-30 minutes until you attract a ride.
It is also very easy to hitch from across the street from the otogar, at least to the top of the hill, saving you a sweltering walk.
The stop above can be used, but an alternative is to walk West. It's a longer walk (1km+) but maybe easier than climbing the steep hill in the sun. From the same central traffic lights, head west past electrical shops on your left; then the Friday market then the new marina. (This marina gets most of the private and private-charter boats, as opposed to the main harbour which handles daily tours, dive boats etc.) After the end of the marina the road joins the main road at a large roundabout; best hitch a little before that, outside the top gate of Can Mocamp. Can is sympathetic to budget travellers - shade, tap water, toilets and fellow travellers available.
There is a 'beach' area a few minutes walk from the centre. there is no actual beach, but there are countless beach loungers which you can use, and this seems to be the typical place to sleep for frugal backpackers. However, User:Damaniacle does not entirely recommend this 'beach', as people come here to drink also, and he was woken up several times during the night, either by curious turks wanting to talk or people copiously vomiting. this may be an isolated experience, or one connected to only the high season. there are other (much less comfortable) places to sleep which are more isolated, for example the marina breakwater.