‘Kingdom of heaven’ is a pious euphemism. It is a circumlocution for ‘kingdom of God,’ used at a time when the word ‘God’ was considered so holy that, out of reverence and awe, it was rarely or never uttered aloud.
The Jewish piety first avoided pronouncing the name of God - Yhwh - aloud, and proceeded to additionally avoid pronouncing the word ‘God’ as well.
In either case, the phrase refers to the followers of Jesus on earth here and now. It is not designation for eternal life in the future. Hence we are told that ‘blessed is ...’ and not ‘blessed will be ...’
The gospels begin with notion that the kingdom is here, not near. Jesus is primarily concerned with the present life.
As with an earlier covenant at Sinai, so also Jesus, establishing a new covenant, is not concerned with the future life, but rather with the here and now. Jesus took care of our eternal future, once and forever, so that we would not worry about it.
We might paraphrase Jesus as saying, “don’t worry about what happens to you after you die. I’ve taken care of that. It’s all set. Focus instead on living this life well: helping others and spreading My good news.”
His message can also be phrased in terms of His kingdom. Jesus makes a clear, although often misunderstood, claim to be a king. He goes on to claim that He reigns: His rule is symbolized in signs and wonders. His subjects are His followers, the community of faith.
Therefore it is understood that the ‘kingdom of heaven’ or the ‘kingdom of God’ are titles for the group. These two phrases refer to the followers of Jesus.
The prayer of a Jesus follower is, “May Your rule be established in my life; Your will be done” (Matthew 6:10), and “seek first to have Jesus as your king” (Mark 12:28).
What does it look like when Jesus is ruling? People loving God by loving each other (Mark 12:28). One gives honor to God by helping other people.
The commandment to ‘honor your mother and father’ (Exodus 20:12) is parallel to the statement that Jesus is a ‘mediator’ between God and humans (I Timothy 2:5): parents have a priestly role, interceding to God on behalf of their children, and presenting God’s Word to their children.
[These thoughts loosely based on a talk given by Dwight Pryor on Saturday, October 01, 1994]