Lt. Hija said:

Now that someone has mentioned it, can somebody explain to me what work of Vader Kylo intends "to finish"?

Vader's last work - we witnessed - was to get rid of the Emperor and convert back to the bright side of the Force...?!?

Well, we witnessed it, all the other characters in the SW galaxy doesn't quite have our benefits though. It's very unlikely that anyone outside of the Rebels, and even then it's just hearsay, would know about it. The Empire would definitely consider it slander and possibly even counter-propaganda. It's possible that Kylo haven't even heard the story, and he definitively wouldn't believe it if he had. As far as he's concerned Vader and the Emperor was killed by Luke and/or got killed by the destruction of the Death Star II. Him looking for Luke could be both to finish the Jedi Purge and to avenge the death of Vader.