My wife and her son are from Dominican Republic. I brought her here on a K1 visa. We got married and now her and her son have their permanent residence card. In one year my wife will hopefully will be a US citizen.

I want to adopt her son (8 years old) so that he has my last name and does not feel alienated in any way from the family. His natural father (Dominican) is unknown and there is no mention of him on the child's birth certificate. What is my best course of action? Thank you.

You need an Attorney - and this is one of those rare cases where there are no other choices.

First of course merely having another last name will not harm a child if treated properly by the parents.

But I can understand wanting to be responsible for the child.

Since this involves out of nation issues the search for the father can be complicated, and you will need to follow local court rules for this. So a local attorney is a have to on these.