There are only 5 references to God's kindness in the New Testament. I was surprised to find it is so few. But they are all big references. Here they are:
Luke 6:35: But love your enemies, and do good, and lend, expecting nothing in return, and your reward will be great, and you will be sons of the Most High, for he is kind to the ungrateful and the evil.
Romans 2:4: Or do you presume on the riches of his kindness and forbearance and patience, not knowing that God's kindness is meant to lead you to repentance?
Romans 11:22: Note then the kindness and the severity of God: severity toward those who have fallen, but God's kindness to you, provided you continue in his kindness. Otherwise you too will be cut off.
Ephesians 2:7: ...so that in the coming ages he might show the immeasurable riches of his grace in kindness toward us in Christ Jesus.
Titus 3:4: But when the goodness and loving kindness of God our Savior appeared, he saved us, not because of works done by us in righteousness, but according to his own mercy
In each case God's kindness adorns, shows off and adds to the richness with which his love and grace are displayed. His love must be great because he is kind to the ungrateful. His grace must be immeasurably great because it is expressed in kindness to the utterly undeserving.
There are 6 references to believers being kind in the New Testament. Here they are:
1 Cor. 13:4: Love is patient and kind; love does not envy or boast; it is not arrogant...
2 Cor. 6:6: but as servants of God we commend ourselves in every way: by great endurance, in afflictions, hardships, calamities, kindness, the Holy Spirit, genuine love
Gal. 5:22: But the fruit of the Spirit is love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, self-control; against such things there is no law.
Eph. 4:32: Be kind to one another, tenderhearted, forgiving one another, as God in Christ forgave you.
Col. 3:12: Put on then, as God's chosen ones, holy and beloved, compassionate hearts, kindness, humility, meekness, and patience, bearing with one another and, if one has a complaint against another, forgiving each other; as the Lord has forgiven you, so you also must forgive
2 Tim. 2:24: And the Lord's servant must not be quarrelsome but kind to everyone, able to teach, patiently enduring evil...
The overwhelming tone here is that believers are kind because they have been shown kindness by God. Kindness is core to the believer's character because it is a primary characteristic of God's grace to us. We have to actively live in it - put it on, take it up, clothe yourself in it. When we do, it commends the gospel and commends real servants of the gospel - people who live and love as if they have been forgiven.
One thought: if a person claims to belong to the God who is gracious and kind, but is not, as a general pattern, kind themselves, it may be that they aren't a believer. I can think of people and churches who say they believe in grace and teach the doctrines of grace and yet themselves are ungracious. I can think of others where an apparent concern for truth is actually a thinly veiled excuse for being hard. Where there is a lack of real life kindness we may well wonder if that person or fellowship is actually living in the grace of God. If we are fogiven, if we are God's chosen ones, we must put on kindness. It is not an optional extra.