Our righteous works have everything to do with our salvation. Read the gospel of John. For example Jesus says those who love me obey my teaching, and those who do not love me do not obey my teaching. John 14.24. What is his teaching, it is based on doing righteous works, that is the message of the gospels. Belief in Jesus comes though righteous works and besides all of this, God said he would reward righteous deeds at the judgement seat of Christ. Naturally when we want to please Jesus, we show it by being righteous.
Our righteous works have nothing to do with our salvation.
Romans 10:8-13
8 But what does it say? “The word is near you, in your mouth and in your heart” (that is, the word of faith which we preach): 9 that if you confess with your mouth the Lord Jesus and believe in your heart that God has raised Him from the dead, you will be saved. 10 For with the heart one believes unto righteousness, and with the mouth confession is made unto salvation. 11 For the Scripture says, “Whoever believes on Him will not be put to shame.” 12 For there is no distinction between Jew and Greek, for the same Lord over all is rich to all who call upon Him. 13 For “whoever calls on the name of the Lord shall be saved.”
Our salvation will produce righteous works.
Ephesians 2:8-10
8 For by grace you have been saved through faith, and that not of yourselves; it is the gift of God, 9 not of works, lest anyone should boast. 10 For we are His workmanship, created in Christ Jesus for good works, which God prepared beforehand that we should walk in them.
You say belief in Jesus comes through righteous works.
Sorry but works are as a result a result of belief.