This Psalm is about righteousness. It is quoted by Paul in Romans to demonstrate that all are in need of salvation from sin, Jews and Gentiles alike.
The writer describes us as “filthy” and Isaiah, perhaps allusion to this Psalm, describes our righteousness as “filthy rags”, a poor covering for us to stand in God’s presence.
Experience bears this out. Not a single day passes without some sin transpiring in our lives, be it a thought, a word, a deed, or an omission. For centuries the Confiteor has been part of the Christian liturgy (http://www.preces-latinae.org/thesaurus/Basics/Confiteor.html).
Acknowledging that we are sinners and that we lack the means to overcome our sin by ourselves is the first step on the road of repentance. Sin comes in a variety of forms from “small” acts of self-indulgence to serious offences that will also result in criminal prosecution if picked up by the secular authorities.
Sin begins in the heart. Jesus taught in the Sermon on the Mount that we have only to think something sinful to have committed sin. We need not act on it to be rendered sinful. We are thus defiled at the root, unfit for God’s heaven, unless He first cleanses us.
Paul, building on the teachings of Jesus, emphasises that we are to focus on love and seek God’s Spirit. This requires self-discipline, but it would be wrong to think that by our effort alone we can overcome sin. Protestantism, looking to Paul’s teachings, advises that we should be conscious of our sin, but keep our focus on living in gratitude for what God has done for us. Where we fall into a deep pit of shame and despair, we are weak against temptation. Faith is, among other things, about trusting God, who is working in us to make us more and more like Jesus. We must focus on Jesus and benevolence and for those grievances in our lives, we must trust that God will make things right on the Last Day.