Vlad: He doesn't have a criminal record, but obviously Luther would easily be able to discover that he moved from Romania and has lived there his entire life - aside from a bit of traveling. His name is pretty well known and respected over there, due to his business & father's reputation.
There's nothing concrete tying him to the murders - he was questioned informally, but a lot of people were and nothing ever came of it - however there are some rumors floating around that he might've had something to do with it. That would require some deeper investigating, though. Also, his reasons for moving to the U.S. are legit - he is in the process of setting up firms in New York.
Cole: Everything is available for him, but it's pretty boring. Grew up in South Boston with a single mother, went to school and became a teacher, moved here. :p
Hannah: Luther probably wouldn't have any trouble digging up her history. Father died when she was seventeen, she attempted suicide when she was eighteen and was subsequently institutionalized for five years, and her stepmother was arrested and convicted on a number of charges, mostly aggravated assault and attempted murder.
Corinne: She did change her last name, but if Luther cared enough to investigate further he'd probably be able to find out that she was raised in an orphanage. Her record is clean, though.