Here is official booking service of Ukrainain railway booking.uz.gov.ua
Official site of Southern Railway where you can check trains/places/prices
Select the following stations: Kyiv, Kharkiv.
There are also private busses Kyiv-Poltava- Kharkiv. Mini busses (marshrootka) are leaving Kharkiv or Kyiv every hour. Duration is about 7 hours. Cost 180 UAH (in fall 2014). They can pick you up at Borispol Airport.
For example, we used this service and recommend them - http://zelenyislon.com/.
Note that private mini-busses are leaving from the following stations:
Prices vary from $50 to $150. Since this is a domestic flight, you need to leave International Airport and turn right. Find Terminal A in 100 meters. Here is schedule of flights:
8.45 - Monday- Friday
9.55 - Saturday
15.20 - Sunday - Friday
21.00 - Sunday-Monday
Dniproavia - 8.45
Austrian Airlines (To Kharkov through Vienna) www.austrian.com
Pegasus (To Kharkov through Istanbul) www.flypgs.com
Wizz Air (To Kiev from London) www.wizzair.com
From Kiev (Zhulyany) to Kharkov www.khors.com.ua