If Christ is in you, although the body be dead because of sin, the spirit is alive because of righteousness. The effect of the sin of Adam and Eve is that there is a duality that characterizes the human person. Among the many metaphors that Paul uses to describe this condition is death and life. While sin subjects the human person to death on a daily basis, in Jesus Christ, the human person is given a new life. The death and resurrection of Jesus Christ has had the effect of countering the effect that sin has on the human person (death). But Paul also uses the metaphor of death and life in rather “constructive” way, so to speak. 
Paul is aware that despite the new life given us in Jesus Christ, our biasness (inclination) towards evil (sin) still resides inside of us and is the cause of our desire to rebel against God and God’s commandments. Paul refers to this aspect of ours as our “lower nature” or the flesh. This lower nature of ours is constantly nudging at us to give in to its lures. The good news for believing Christians is that they do have alternatives. Believing Christians can choose not to act on the nudging of their lower nature because in their participation in the life of Jesus, they have put to death those desires. But that is not all. After putting to death their lower nature, believing Christians take on a new nature, that of life. St. Paul refers to it as being alive in the spirit. By spirit, St. Paul refers to that mark of God that we acquire at creation (the spirit/breath of God). It is this spirit of God that is destroyed when we sin. As such when Paul talks about believing Christians being alive in the spirit, all that he means is that the mark of God is awakened in us and we are enabled once again to live full lives as sons and daughters of God (as God had intended it to be). Such a life is characterized by remaining in the relationship that God has created for us (righteousness=right relationship).