Oh, wow. This one gets debated quite a bit. The question becomes, in what way might the righteousness of the law be fulfilled in us? Through faith or through actions? I take the latter, as that seems to fit the context better. If you look closely, what Paul says God made possible for us to fulfill here was
the righteousness of the law, not the righteousness of faith. By that, he meant what he was referring to earlier in Chapter 2, when he talked about the righteousness the believing Gentiles were now walking in:
14 for when Gentiles, who do not have the law, do by nature the things contained in the law, these, although not having the law, are a law to themselves, 15 who show the work of the law written in their hearts, their conscience also bearing witness, and between themselves their thoughts accusing or else excusing them) 16 in the day when God will judge the secrets of men by Jesus Christ, according to my gospel.
Paul was not talking here about Jewish ceremonial law but the two greatest commandments, upon which the entire law and the prophets were based: To love the Lord with one's whole heart, mind, soul and strength, and love one's brother as oneself.
So I take the passage in Romans 8 to be saying the same thing Jesus said about the Pharisees. They did not keep the heart of the law. Although they appeared to keep it externally, in many ways they were actually wanting to break it in their hearts, and this is why Jesus said to the disciples, "Unless your righteousness exceeds that of the Pharisees, you shall in no wise enter into the kingdom of Heaven." He was speaking again of practical righteousness. He meant that we must not be hypocrites (i.e. "actors") like the Pharisees were, many of whom put on a religious act but deep down did not truly love God nor their fellow man. As for us, however, the teaching was that true believers in Christ
fulfilled the righteousness of the law. It is what the Spirit was poured out for; that we might walk not after the flesh but after the Spirit, and fulfill the two greatest commandments rather than merely pretending to do so just to look religiously devout.
Those are my thoughts. Not everyone shares them with me. I'm glad Marks isn't here; this post would possibly start a ten page, seemingly endless debate, Lol.