The most simple definition for Righteousness is to be in right standing with God.
Any sin during our lives has made a division between ourselves and God, and any sin is unrighteousness, implying that after sinning there is something that is between ourselves and God causing a division. This is why Adam after sin went and hid from God. Gen 3:8 (and the man and his wife hid themselves from the presence of the Lord) , but God was looking for him..
Gen 3:9 (But the Lord God called to the man and said to him, “Where are you?)
Herein is the key to the gospel. Most people try to stop sinning in order to be reconciled to God, but its God reconciling us to Himself that is the key to the Gospel. God has
5 :19 …in Christ God was reconciling the world to himself, not countingtheir trespasses against them, and entrusting to us the message ofreconciliation.
How then can we be reconciled with God if sin will always cause some division? That is why God has made us Righteous, forgiving our trespasses, in order that we can be reconciled to Him and once reconciled does it then make sense to continue in sin, most definitely not.
Rom 6: 1 What shall we say then? Are we to continue in sin that grace may abound? 2 By no means!
How do we achieve this? By Faith. Come to the realization what Jesus has come to do, reconciling us to Himself by taking away the things that brought division, and instead made a way. Herein He has made you righteous, now walk in that righteousness, rooted in faith, and believe that you are now the righteousness of God and walk therein and soon you will be acting accordingly.
But what if I fall? There is grace, appeal to Gods mercy, to get grace, not to sin, but to stop sinning. In doing this you will soon find that that desire to sin will also be removed by the Blood of the Lamb.
2 Cor 5:20 We implore you on behalf of Christ, be reconciled to God. 21For our sake he made Him to be sin who knew no sin, so that in Him we might become the righteousness of God.