There is an official process for picking a child to adopt in Ukraine and that is to visit the SDA offices in Kiev and view portfolios of kids that they provide to you, and then choose one.
That is, unless you already have a referral for one. It is a little mysterious to me how that works officially, since it is illegal for a third party to act as an agent or broker in the adoption process. How do you find out about a child other than the SDA? Word of mouth. Hosting. Prior visits to orphanages. Siblings. Providence. Good luck. We found out about Sergei and Valya in those ways.
This is on my mind today, because I was reading some of the adoption blogs that we have links to from ours, and found some cases of failed Ukrainian adoptions. Several families recently went to Kiev to adopt, and the only kids that the SDA would show the families had serious medical problems. They had gone to Ukraine without a specific child in mind beforehand. I will let you read about them if you wish, but it is just too sad.