That one is specific in that kills must be kills, not Takedowns. They must be your kills, you get the last hit / execution, not an assist. You will get the kill streak message each time it is genuinely your kill, and you need to get to Godlike without dying. Against players you have almost no chance, against AI it is not so bad. It is the hardest Order in the game.
If it's a bug with the orders please report it to support so we can look into it and ensure that order rewards are going out as they should.