Both of these words are from the same Greek word.  It helps if we can understand that righteousness is a right relationship; it is a right relationship with God, our fellow humans and the covenant.  Justification has been misunderstood because we have made it a legal or forensic word.  When we come to Christ in faith we are justified, which means that we are set right with God through Him.  We are set right to live right (righteous lives).  This involves the good works of compassion, mercy and justice.  To say it another way, we are set right to do right and be right.  We have been justified (set right) to live lives of righteousness. 

“By entering through faith into what God has always wanted to do for us—set us right with him, make us fit for him—we have it all together with God because of our Master Jesus. And that’s not all: We throw open our doors to God and discover at the same moment that he has already thrown open his door to us. We find ourselves standing where we always hoped we might stand—out in the wide open spaces of God’s grace and glory, standing tall and shouting our praise. (Romans 5:1–2 The Message).