What steps can be taken to find any living relative of a employee that has died and their is no one listed as the next of kin?

There are 'heir search' services such as "International Genealogical Search" available if you are stuck ... I'd try that if you have no other option. Generally coroners handle disposal of remains however -- why do you feel this is your responsibility? The final paycheck or other property in your possession may need to go to heirs and if there are no heirs to the State, so this may still be a concern but you need to insure you're not overstepping your authority.

You don't say where the employee last resided. However, the local governments will have a process by which to deal with unclaimed bodies and the possessions of decedents. I would suggest you call the police to find out what office that may be in your locality. (In NYC we have the office of the Public Administrator.)You are not obligated to find the next of kin and only a governmental official could legally enter the decedent's apartment or house (at this point) to search for information about the next of kin. If they find information they will pass it on to the next of kin.