2He that walketh uprightly, and worketh righteousness, and speaketh the truth in his heart." (Psalm 15: 1-2)
Under the Old Covenant, righteousness was defined by the Israelites' adherence to the Mosaic law, which no one could keep:
"And Joshua said unto the people, Ye cannot serve the LORD: for he is an holy God; he is a jealous God; he will not forgive your transgressions nor your sins." (Joshua 24: 19)
Joshua had to declare plainly to the Israelites - they could not keep God's law, and thus of necessity sacrifices were ordained (Exodus 20: 24-26)
Paul makes it more plain, but with the promise of the Savior now fulfilled for us in Christ:
"Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God." (Romans 3: 19)
"For all have sinned, and come short of the glory of God;" (Romans 3: 23)
So, how can we walk uprightly today, since none of us can keep the Mosaic law?
Paul relates what it means to walk uprightly today, under the New Covenant:
"But when I saw that they walked not uprightly according to the truth of the gospel, I said unto Peter before them all, If thou, being a Jew, livest after the manner of Gentiles, and not as do the Jews, why compellest thou the Gentiles to live as do the Jews?" (Galatians 2: 14)
Today, under the New Covenant, to "walk uprightly" means that we believe in Jesus' Finished Work, that He has put away all our shame, all our sins, that we are no longer under law, but under grace:
"For sin shall not have dominion over you: for ye are not under the law, but under grace." (Romans 6: 14)
"For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people:
11And they shall not teach every man his neighbour, and every man his brother, saying, Know the Lord: for all shall know me, from the least to the greatest.
12For I will be merciful to their unrighteousness, and their sins and their iniquities will I remember no more." (Hebrews 8: 10-12)
We abide if we walk uprightly, and our walk is all based on believing in Jesus and His Finished Work.