Where is the check-in counter?

The location will vary depending on your airline. For details, please check the flight board inside the airport or "Today's Flight Information" on Kansai Airport's official website.
[Domestic flights] 2nd floor of Terminal 1 (ANA, JAL, Jetstar, StarFlyer) or Terminal 2 (Peach) 
[International flights] 4th floor of Terminal 1 or Terminal 2