The UEFA Football Cup will be visited by up to 1 million national teams supporters. One of the easiest way to get to different cities in Ukraine is to go by Kiev(Kyiv). It's clear that a problem to get to Kharkiv, Lviv, Donetsk will not be easy to solve as the number of tourists in June will raise 1000 times and all the plane and railway tickets will be sold out.
The only way in this situation is a taxi transfer (car or minibus) from Kyiv Airport. Your taxi driver will meet you in the building of Kyiv Boryspil Airportand the car will leave for the city you need immediately after you're in the car. So, it's the fastest way to get from Boryspil Kyiv Airport to Kharkov,Lviv or Donetsk.