I travel to KBP (Kiev) quite often, and I can offer a few sugegstions. First, the International Airport (KBP) is located approx. 30 minutes outside of Kiev. If you are going to travel domestically, you will need to transfer to the domestic airport. Ukraine has changed some of its VISA requirements for US citizens (thank God.) They made it so difficult to get into that country, now it is a lot easier. You will also be required to purchase health insurance at the airport before you will be allowed to proceed through immigration. It is not that expensive, but it is another way the government extorts money from people. Also, be aware of the currency and article declaration regulations. BE VERY PRECISE, or you may end up in prison and/or your possessions will be confiscated. List all your articles of value (watch, rings, computer, accessories, etc) and declare ALL your money (if over $1000USD.) Failure to do this will cost you dearly. Count it exactly, you may be asked to count it in front of a customs officer, and if you are off, they can legally take your money or worse!
Also, be prepared to pay a bribe in order to leave the country. This is so common it is pathetic! What will happen is something like this.....As you arrive into the airport, your passport will be asked for by a local policeman. He will take it and examine it for your registration stamp (not VISA stamp, but your registration Stamp.) You get this if you stay in a hotel, and they will take your passport and have it stamped for you. If you are staying with a private individual or flat, then you must register within 3 days of arrival at the provincial police station. If you do have this stamp, he will detain you in mindless ways so as to make you late for going through bag X-Ray and checkin. You will have very little time to do this, then it becomes appearent what he wants...$20.00 USD. You pay him, and he lets you through. This at times happens a couple of times. It is wrong, but the police are corrupt! They are paid very little, and they use this as a means to survive.
The domestic airport is not very nice, but passable.
I hope this helps.