The key to understanding Romans 6-8 is to understand the clear "before and after" Paul presents repeatedly throughout the section. The first is in 6:17-18:
"But thanks be to God that, though you used to be slaves to sin, you wholeheartedly obeyed the form of teaching to which you were entrusted. You have been set free from sin and have become slaves to righteousness."
Notice the clear timing here. The Romans used to be slaves to sin before they believed. Now they are free from sin and have become slaves to righteousness. Paul gives no middle ground. A person is either a slave to righteousness or a slave to sin. You are not both at the same time. The timing for Paul is not ambiguous. Before you commit to Christ you are a slave to sin. After you believe, you are not to "let sin reign in your mortal body so that you obey its evil desires" (6:12).
Paul gives the same contrast in 6:19-22 and then again in 7:5-6. His statement in Romans 7:5-6 is worth quoting because this statement is in the lead up to what he will say later in that chapter:
"For while we were in the flesh, the sinful passions, which were aroused by the Law, were at work in the members of our body to bear fruit for death. But now we have been released from the Law, having died to that by which we were bound, so that we serve in newness of the Spirit" (NASB).
Again, the timing is clear. Being "in the flesh" is a matter of a believer's past. And the "fruit for death" that Paul has in mind is also clear from the previous chapter. 6:19 puts it clearly: "you used to offer the parts of your body in slavery to impurity and to ever-increasing wickedness, so now offer them in slavery to righteousness leading to holiness." And when Paul says we have "been released from the Law," is not talking about some legal hocus pocus where God treats us like we have become righteous even though we continue sinning as before. You would have to ignore everything he has said in the previous chapter to read him this way.
No, there is not the slightest ambiguity here. Paul is saying that we were slaves to sin. We used to act in ways that lead to death. Paul is saying believers no longer let their sinful passions control their actions. Believers are set free from the law of sin.
Romans 8 confirms this same train of thought yet one more time: "the law of the Spirit of life in Christ Jesus has set you free from the law of sin and of death" ... "so that the requirement of the Law might be fulfilled in us who do not walk according to the flesh but according to the Spirit" (8:1, 4). "Walking" is the Jewish word for how one lives (halakah), confirming that Paul is not talking about something only figuratively fulfilled in us. He is talking about believers actually keeping the essence of the Law, like the Gentiles he mentioned in Romans 2:15 who demonstrate the Law written on their hearts. 
Paul's overall train of thought in Romans 6-8 is thus quite clear. One is either a slave to sin or a slave to righteousness. Before the Spirit, a person is a slave to sin with the result that they act sinfully and follow the lead of the sinful passions inside them. However, after the Spirit, a person is set free from the law of sin, a person is freed from sin and becomes a slave to righteousness, with the result that they act righteously and actually fulfill the righteous requirement of the Jewish Law.
It is only when we have this context firmly in mind that we should approach Romans 7:14-25. The popular interpretation of these verses is so deeply engrained in the contemporary Christian's mind that it will take every effort to keep most readers even at this point from chucking Paul's clear train of thought out the window. Paul cannot be saying that he is still a slave to sin or that he continues to be unable to keep the heart of the Jewish Law unless he is saying he is not a believer.
"We know that the law is spiritual; but I am unspiritual, sold as a slave to sin" (7:14). Paul cannot be talking about himself at that very moment because he has said clearly, repeatedly, and emphatically that believers used to be slaves to sin but are now free (6:17-18; 19; 20-22; 7:5-6; 8:1-4). If Paul is talking about the default state of the Christian or of himself, then his train of thought here is fundamentally incoherent. The only way to read Romans 7 this way is to rip it completely from the rest of Romans!
The only way to read Romans 7:14-25 coherently is to see Paul putting himself into the shoes of the default Jew who has not yet believed, not yet been baptized into Christ, and has not yet received the Spirit.  It is a person who "wants to do good" (7:21, NASB) but is incapable because s/he is a slave to sin. In short, Paul is not talking about his current experience but he is explaining what the purpose of the Law was even though unbelieving Jews are unable to keep it even in its essence. 
Paul's theology would have starkly raised the question of the Law among Jews. Paul is saying that keeping the Jewish Law does not make a Jew right with God. Only the death of Christ and faith in what God has done through it can make a person right with God. Well then what was the purpose of the Law, then, Paul? He states clearly in Romans and Galatians that its purpose was to show me my need for Christ (e.g., Gal. 3:24). Romans 7:14-25 presents this need dramatically.
I am a Jew. I want to keep the Jewish Law. The Jewish Law tells me what God desires of me (meaning those core parts that apply universally). But I find in my default state that I am unable to keep it. I discover that I am a slave to sin, that I am "utterly sinful" (7:13). Who will free me from this body that makes me do things that lead to death?
"Thanks be to God—through Jesus Christ our Lord!" (Rom. 7:25). That is to say, when I am incorporated into Christ, I am set free from this horrible default state in which humanity finds itself, including Jews. We have seen this exuberant exclamation before, "Thanks be to God." It was back in Romans 6:17, and it is worth quoting those verses all over again to show that Paul's train of thought here is exactly the same as it was there: "thanks be to God that, though you used to be slaves to sin, you wholeheartedly obeyed the form of teaching to which you were entrusted. You have been set free from sin and have become slaves to righteousness." 
Romans 8 thus continues with the believer, returning to much the same point Paul reached at the beginning of Romans 5. Paul had reached the point of talking about peace with God in Romans 5:1 after his lengthly discussion of the need and basis for justification in the first four chapters. Romans 8:1 returns to this point of "no condemnation" after over two chapters talking about the problem and overcoming of Sin in a person's life. The rest of Romans 8 will celebrate the real benefits of the Spirit's power in the believer, both now and in the redemption that is to come.
 It thus seems very likely that Paul had Gentile believers in mind in Romans 2:15. Therefore, in Romans 8:4, Paul is not thinking that the righteous requirement is literally fulfilled in Christ and only figuratively then fulfilled in believers. Paul is saying that the law of the Spirit literally empowers believers to keep the heart of the Jewish Law, what he calls "Christ's law" in 1 Corinthians 9:21).
 The suggestion that Paul has Adam in mind in 7:9 seems to overread the passage. Paul is presenting a logical, not an autobiographical or chronological sequence.
 Paul's claim here is difficult because Jews in general did believe they could keep the Law appropriately to God's expectations. Some have accused Paul of misrepresenting the Law or even of misunderstanding it. Nevertheless, we do find this same sentiment elsewhere in the New Testament (e.g., Acts 15:10) and Paul implies the same in Galatians 6:13.
 From the standpoint of our understanding, it is unfortunate that Paul went on in the rest of Romans 7:25 to summarize the state of his hypothetical unbelieving Jew wanting to keep the Law--"I myself in my mind am a slave to God’s law, but in the sinful nature a slave to the law of sin." Just this restatement is enough for many to ignore the victory his train of thought has just pronounced through Christ.