I just came back from Bali and during this trip I visited Kelingking beach also. 1st you have to take speedboat from Bali to Nusa Penida ( costs around 1500 INR, including transfer from Bali hotel to jetty). Then take a bike or car on Nusa Penida jetty for KelingKing beach. The Bike costs around 400-500 Rs and car around 2500-3000 Rs for full day. The roads are pretty bad and it's an adventure riding a bike there. I have filmed my entire trip and posting weekly videos on Youtube. If you are interested, you can follow me by clicking this link
https://www.youtube.com/channel/wanderingmonk. Hope it helps.
The easiest way to explore the island is by scooter. They cost around 70,000 IDR per day, and can be hired directly from the harbour port. If you're on a private Nusa Penida tour then Kelingking Beach will usually be the first place your tour guide stops.