We use that phrase in American culture to express our displeasure with someone. We simply announce that "he's dead to me" and then proceed to ignore or downplay anything that person does. It's an amplification of the junior high tactic of not speaking to someone.
There was, in times past, a very real sense in which people treated the living as dead as a form of passing judgment on them. People were treated as if dead for marrying badly, choosing certain professions their family found distasteful, or even taking the wrong side in politics. It's no way to treat another human being. Really and truly.
Nothing in Scripture allows Christians to treat other people that way. However, we are to act as if we are dead to certain things. It's the opposite of our cultural behavior where we expect others to do the dying for us. We are the ones that need to choose to die.
One of the things we need to die to is sin. That's what Paul is referring to in Romans 6:1-4. Sin will not roll over and die for us. There's really no way we can kill it ourselves, but we do have the option to die ourselves. Fortunately, we don't have to literally die, but instead that we are to identify with Christ in His death.
You see, we can't honestly expect for the temptations of this world to leave us alone. We can't expect the old habits of our sinful nature to leave us alone. What we have to do is be so dead as not to heed them. The difficulty, though, is that often we try to die to sin without being alive to something else. It's a way that our materialistic philosophy has penetrated our Christian behavior. (note: we're not talking Madonna Materialism here. Philosophically, materialism isn't about getting more stuff, it's that the stuff is all there is, that if something is unseen or unquantifiable, it doesn't exist. This leads to the behavior of materialism where, since stuff is all there is, we try to get lots o'stuff.) Materialism as a philosophy holds that after death, there's nothing.
Biblical philosophy (for that matter, most religious philosophy) teaches us the opposite: that there is much more after death than we see here. Death is the ending of one thing, but the beginning of another. We Christians believe that physical death is just the beginning of eternity. Yet we try and "die to sin" without living to something else.
We've got to learn to live after our death. After we die to sin, we have to learn to live to God's glory. All of our pursuits, all of our energy, time, finances should be focused in that direction. Is it?
So many of us try to live to both sin and to Christ, and yet it's clearly impossible to do so. This is the source of our frustration as followers of Christ: we have to choose, which world do we want to live in? We need to choose to go ahead and live as dead to this world.