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The word "righteousness" shows up in Paul's letter to the Romans about 35 times (various forms of the word, like "right," "justice," and "justification," increase that number). Obviously, Paul has a lot to say in Romans about righteousness. But have you ever considered what righteousness actually is?
In Romans 10:5, Paul states, "For Moses writes that the man who practices the righteousness which is based on the law shall live by that righteousness" (NASB). As far as Paul is concerned, this is a statement of fact and demonstration: based on his indirect reference to Leviticus 18:5, everyone can see and everyone knows that the only possibility for those who want to try to be righteous on their own is to live by the laws that help show us that God is righteous. If this is all that Paul had said on the matter, it would be entirely reasonable to conclude that his definition of righteousness is a set of laws that teach us how to live, which are modeled after God's own righteousness. But Paul goes on to talk about how God has shown us his righteousness by actually giving it to us.
In Romans 10:6-8, Paul quotes the Greek translation of Deuteronomy 30:12-14 to make the point that you don't need to search the heavens or the ocean's deep abyss in order to find righteousness; instead, that righteousness is already close to you and me. And in answer to the question of HOW God's righteousness is close to us, Paul answers in Romans 10:9, "That if you confess with your mouth Jesus as Lord, and believe in your heart that God raised Him from the dead, you will be saved" (NASB). Paul's point in Romans 10:5-9 is twofold: Jesus IS God's righteousness, and you get God's righteousness by putting your faith in Jesus.
Friends, God has already shown us what his righteousness is - and it's Jesus! And God has already given us his righteousness by giving us Jesus - and when you have faith in Jesus, God makes you righteous! Be encouraged by the righteous character of God as found in the Lord Jesus! .