It is the Japan Rail (JR) Haruka. The Haruka runs about twice per hour.

The JR Station at KIX is on the second floor and across the pedestrian walkway over the main airport driveway. There are numerous signs in the Arrivals Hall in multiple languages which direct you to "Rail Transportation." Just follow the signs up the escalator and across the pedestrian bridge. It is difficult to miss the way. The JR ticket office is in a green-trimmed, glass walled space that is well marked. Any ticket machines will be right near-by. If you are unsure, it might be simpler to buy your ticket from an agent if the line is not long. There are many English-speaking agents.

At Kyoto Station, the JR ticket office is right inside the main entrance on the Karasuma Dori side. It is also in a glass-walled space with green trim. Ticket machines are next to the office near where the queues start. JR offices all look pretty much alike.

Example -



The best tool for planning rail travel is HyperDia (link below.) For destinations, enter "Kyoto" or "Kansai Airport." It will return many less convenient options unless you select "More Options" and deselect everything except "Limited Express."

http://www.hyperdia.com/en/#