Righteousness  vs  Self-righteousness
Righteousness = Right in God's sight.
  Example:  When a judge declares a person "not guilty", that person is "right in the judge's sight".

Self-righteousness = "right in my oun sight"
  Example:  When I think I am right and declare that I am "not guilty", I am self-righteousness.
Righteousness
To be righteousness, be in the right, be justified, be just.  It is a legal term which involves the whole process of justice.  It is a legal term signifying justice in conformity with the legal corpus (the law).

When man sinned, he, man turned his face away from God.  And God turned around because He could not look upon sin.

Justice demands death.   
So God had to redeem man without going against His own righteousness.  He had to come up with a plan.  In that plan someone had to die.  And that is what Jesus did.    He died.  In that death, God turned around.  So, now man has to turn around by excepting what Jesus did.  For we only can go to God through the righteousness of Jesus.  We can never do that in our own righteousness.  The scripture say our righteousness is as filthy rags.

Righteousness is rightness
Rightness is conforming to a standard.  And the standard is the character of God Himself.  Righteousness of God is - God's conformity of His own attributes.

The attributes of God are those things that we can contribute to God.  He has devised a way by which He can provide for sinful man.  His righteousness that conforms to His standard, which is the righteousness of Christ.