Self-righteous according to dictionary.com is to be confident of one’s own righteousness, especially when smugly moralistic and intolerant of the opinions and behavior of others.
This is the worldly definition of self-righteous, or at least one of them which is a harsher than some others. The word smug is used in this one and is missing in other sources. Intolerant is also used in this definition and absent in others. A strong abrasive definition often brings home strong resentments to the reader, as it should. No one likes to be called self-righteous.
What is missing in all definitions is the one definition required to complete our understanding. It assumes we all have the same understanding of what the word righteous means. In worldly terms and understanding it means to be right. That thought falls short in what the spirit has to say on the matter.
Philippians 3:8-10 English Standard Version
8 Indeed, I count everything as loss because of the surpassing worth of knowing Christ Jesus my Lord. For his sake I have suffered the loss of all things and count them as rubbish, in order that I may gain Christ 9 and be found in him, not having a righteousness of my own that comes from the law, but that which comes through faith in Christ, the righteousness from God that depends on faith— 10 that I may know him and the power of his resurrection, and may share his sufferings, becoming like him in his death,
The Blue Letter Bible renders this bible usage of the term righteous in a broad sense: state of him who is as he ought to be, righteousness, the condition acceptable to God, the doctrine concerning the way in which man may attain a state approved of God.
The way that we may stand upright before God is to be in Christ by faith. That is righteousness.