If my memory is correct, the first words from Rick Warren's A Purpose Driven Life are "It's not about you." These would be apt words to open up Wright's 6th chapter of What Saint Paul Really Said on the topic of the righteousness of God.
Righteousness is not about you. Righteousness is about God. It means that God is faithful and trustworthy. It means that God has integrity. It means that when God makes a promise, God will back up those words with action. This is "Good News for Israel" as the chapter is entitled. It means that God will act to redeem Israel from her exile. It means that Israel will be saved and redeemed from the hands of her oppressor. It means that evil will be defeated. And it means all this because God is righteous.
We could restate this according to the understanding that righteousness means "right-relationships" according to the terms of the covenant between two parties. God has established his covenant with Israel, therefore, for God to be "righteous" means that he acts according to the terms of the covenant.
Within the span of this short, 15-page chapter, Wright unpacks the law-court setting in which "righteousness" language is derived as well as looks at key passages from Paul's letters that capture Paul's thinking on this topic. It would be tempting to quote a lot more of Wright's words from the chapter to illustrate what he is talking about, but Warren's phrase, as applied to righteousness, seems sufficient; "It's not about you." It's not about a certain moral quality that people are given. It's not about any kind of slight of hand by which God no longer sees human wickedness and idolatry. Rather, righteousness is a attribute and characteristic of God. It's is who God is. God is righteous.
This has tremendous pastoral implications because it provides the assurance that God can be trusted to do the things God said he will do. God may not do them in the ways his people expect; and certainly a suffering servant messiah was not expected. So yes, we must be prepared for God to act outside of our human-minded categories. But when God says he will redeem his people, he will. When God says he will forgive sins, he will. When God says that the lion and the lamb will lie down together, they will. When God says death has lost its sting, it has. And not only will this things happen one day in the future, but in the Christ Jesus these signs mark that the reign of God have already begun.
And none of this is because I've become good, or because God sees me through "blood stained glasses" or because God simply chooses to ignore my sin. It is all because God is righteous and acts out of right relationships toward sinful and broken humanity because of God's great covenant of love for all of creation.
This is a God I can worship. This is a God I can draw near to. Not because I've somehow become good in the moral or the forensic sense (though certainly I do strive for obedience), but because God is good.
"No one is good but God alone," Luke 18:19. And that is why the righteousness of God is good news for Israel and all of humanity.