Local GA airfield near Kiev is called
CHAIKA, which literally means
SEAGULL 
So, SEAGULL61, looks like you simply must go and fly there
Last year there was an active movement in Ukraine towards becoming JAA full member state. Last news I've read in Ukrainian press is that it should become true some time in December; however, due to recent political changes in Ukraine that process might be "postponed".
If you want to organise a flight by your own, it might be a hassle because of buerocracy (e.g. permission request to military/air-defense is still required ... ). I would recommend you to contact aviation authority of Ukraine for help.
Another paradox : as far as I know, AVGAS is not produced in Ukraine and it's import/sales are presumably not regulated by the law. If that's true, then that is somewhat crazy. So I would be careful about AVGAS there.
To find out more, you can probably check this forum :
http://www.aisukraine.net/newforum/index.php
Some authority employees are ocassionally replying the posts from curious VFR pilots.
Another option is to mail in English to aircompany at avionika.kiev.ua who is administrator of Aeroclub web-site :
http://www.aeroclub.kiev.ua/ That website contains very good and up to date information about GA flights in airspace of Ukraine, including documents and procedures; but it's all in Russian.... Admin will probably help you with useful info.
Never flown in Ukraine, curious myself and would be interested in hearing about any experience.