John 13:1 "Now before the Feast of the Passover, Jesus knowing that His hour had come that He would depart out of this world to the Father, having loved His own who were in the world, He loved them to the end."
"He loved them to the end"
Jesus didn't just love His disciples until the end of His life, He loved them to the - 'end of His love.'  In other words, there's nothing He or the Father held back in their love from us.
Romans 8:32 "He who did not spare His own Son, but delivered Him over for us all, how will He not also with Him freely give us all things?"
When a heart has love in it for someone else, it never wants to stop doing more and doing better for the other person.  And that's exactly what God did for us - gave ALL that HE has.
And in return, I believe the mark of a person that really loves God is two-fold:
1. They love in action (they are do-ers) of God's Word (1 John 3:18, John 14:15)
2. They always want to love the Lord better, and hence, they cleanse themselves (1 Jn 3:3)
The proof is the same in all relationships.  For example, imagine a husband and wife who love each other.  What is the proof of their love?  The proof is that they are trying to love the other one better!  If they can further the good of the other person somehow (even at cost to themselves) - that is love.  "How can I be a better husband to my wife?  How can I be a better wife to my husband?  Where can I improve?"
I believe this is why the Bible often talks about "Cleansing ourselves."  It's making ourselves more fit to do good toward God, the One we love.
2 Corinthians 7:1 "Therefore, having these promises, beloved, let us cleanse ourselves from all defilement of flesh and spirit, perfecting holiness in the fear of God."
In the new covenant, obedience and holiness are not mainly about the 10 commandments (9 of which are regarding EXTERNAL things).  Obedience to God's commands is primarily regarding inward attitudes of the heart.  We can see that clearly when Jesus preached the sermon on the mount.  With that one sermon, He changed the whole emphasis of what God's people should live for - external purity only, to internal purity.  And to set ourselves to go to higher and higher levels of that purity, like God Himself is pure.
Imagine how heavenly a marriage would be where both husband and wife ask each other regularly, "Darling, how can I love you better?  What can I do better today than I did yesterday?"
A parent who loves their children will also have the same attitude.  They will ask themselves continually, "What can I do better?  Am I teaching them the right ways of God?  Am I setting my children up for a good future?  Am I encouraging them?  Do I have the right balance of grace and truth; not too strict and not too light?  Have I been patient with them?"
And if the goal is to look at ourselves honestly IN TRUTH, and  to take advice and correction (whether from God's Word, or our conscience, or others) on how we can do better toward the good of another; that is exactly the same as what it means to walk before God, carrying our cross daily and cleansing ourselves.
We are carrying our cross to follow Jesus and to love and be in fellowship with God.  And so the main thing we do is not to write love letters (saying fancy prayers), or keep a clean physical house (acts of service)... the main thing is: how do I love better?
When we love someone, we also have a joy in their good.  Just like the joy we get when good comes to ourselves.
I was searching for a photo to associate with this blog.  When I typed in the word "Love" into the image search, almost all of the images are of two people hugging each other, or holding each other's hands.  I think there's a lot missing from those pictures of love - it's similar to how Judas kissed Jesus.  Instead, when I type in the word, "Support" the image above comes up (a hand holding up a tree).  I believe that's a much more accurate picture of what love is - a consistent commitment to the good of another, even at our own cost.  Because that's what God has done for us.
Let us have a constant cry in our heart - "Lord help me to love better", so that it translates to action.
1 John 3:18 "Little children, let us not love with word or with tongue, but in deed and truth."