There are two types of righteousness;
Self righteousness(of Man) and the Righteousness of Faith (of God)

Isaiah described Self-righteousness in verse 64:6 saying "it appears as a filthy rag before God"

The Pharisees are a classic example of self righteousness, they portray the "holier than thou" attitude that stinks before the LORD. No wonder he called them snakes, brood of viper's, blind fools and so on. His harshest words were preserved for the self Righteous folks and not the sinners, prostitute and tax collector's.

Paul clearly understood this, formerly known as  "the Pharisee of Pharisees". This Chief Pharisee had an encounter with the Lord and was transformed.  He wrote saying "And be found in him, not having mine own righteousness, which is of the law, but that which is through the faith of Christ, the righteousness which is of God by faith" Phil 3:9

The veil over him was torn, now he could see.

"Not having my own righteousness" 

Now he knows he can't afford to bring his own righteousness before God.
First he said "and be found him" Only when we are found in Christ can we be Righteous.

Christ has been made for us Wisdom, righteousness, Holiness and redemption.

In Him we have redemption through his blood, the forgiveness of sins, according to the riches of his grace. Eph 1:7

The phrase "In Him" makes all the difference for you and I.

Our Salvation and Righteousness is according to the richness of his Grace, not according to your performance or law keeping.
How rich is his Grace? Richer than what u can imagine.

Paul saw something.

He said in Romans 1:16 that he is not ashmed of this Gospel (Grace), that it is God's power unto salvation.

The world is fallen, the people likewise, if nothing is done, they'll all end up in hell.
But God wants to save the world and His power to save the people is in the Gospel.

Grace is the Gospel and its all about Jesus. The Gospel was intended to reveal something to us.
After saying he is not ashamed of the Gospel, the next verse.
Romans 1:17
For in the Gospel the righteousness of God is revealed—a righteousness that is by faith from first to last, just as it is written: “The righteous will live by faith.” NIV

The purpose of the Gospel is to a reveal The God kind of righteousness, a righteousness given to us as a free gift.
A righteousness that comes independent from the works we do.

Isaiah, the old testament prophet knew something about this righteousness and said

Isaiah 45: 24; "Only in the Lord shall one say I have righteousness"

Isaiah 53:11; " He justifies many and makes many righteous"

Isaiah 51:6; "The world may end, but His righteousness and salvation is forever"

We have received an everlasting righteousness from the LORD.
Hallelujah!

When we believe in Christ and believe we have been made Righteous, that's perfect obedience.
New testament obedience is Obedience to the Faith not obedience to the law.
And its emphasis is on right believing not right doing.
Right believing has proved to produce right living.
We must believe rightly and be established in righteousness.