The Kingdom of Heaven,  what is it, exactly?

These are the 31 instances of the term 'Kingdom of Heaven' and all spoken by Jesus, and only in the gosple of Matthew.

Jesus taught on this kingdom, before He went to the cross. . Since Jesus came down from heaven, and ascended back up to heaven, He was the representative of heaven, of which God reigned.
Jesus often said the Kingdom of Heaven is at hand, meaning He and what He was going to accomplish for us. 
 

After Jesus finished the work of God on the cross, he suffered through grace for all of us, and now we enter the covenent of grace, being the Kingdom of Heaven. . and that being a state of the believer, in present time. .it could be compared to what Paul said here:
 

You will notice that below verses Matt:13:24,31, and others, it begins small and grows, with our understanding, knowledge and with our Lord.

Notice Matt.23:13, Jesus rebukes the Jewish religious leaders, who are rejecting Him, and holding back the possibility of the Jews entering into the Kingdom of Heaven, coming after Jesus work on the cross was finished.  It is the state of living in the presence of not only belief in Jesus, but growing in the peace of His fellowship, it is His covering of righteous in our lives, it is the freedom from the bondage of sin, it is the covering of his sanctification of our lives, and all goodness that the Lord affords to us through all His blessings.

Is the Kingdom eternal. . yes, of course, it continues into eternal life.  It starts here on this earth, and goes on eternally, into the next life, but Jesus never mentions it as an eternal state, but all of his parables indicate it is a place of spiritual awareness in Christ, that grows in us, from the moment of our salvation.

Jesus also uses the term 'Kingdom of God' interchangably, the only difference is that God is more specifically the source of this heavenly kingdom.   Jesus, filled with the  Holy Spirit of God, was here to teach us of the coming period of grace, that Jesus paid for on the cross. 

The first term, a little more general, the second term, and little more specific, same kingdom.

This term 'Kingdom of God' used by Jesus in all the gosples.  Paul used this term many times in his teachings as well. 
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There are 69 instances of the use of 'Kingdom of God' in the NT. more below on the Kingdom of God.
 
 

Matthew 3:2  And saying, Repent ye: for the kingdom of heaven is at hand. (coming after the cross)

Matthew 4:17  From that time Jesus began to preach, and to say, Repent: for the kingdom of heaven is at hand.  (again, after the cross)

Matthew 5:3  ¶Blessed are the poor in spirit: for theirs is the kingdom of heaven.

Matthew 5:10  Blessed are they which are persecuted for righteousness' sake: for theirs is the kingdom of heaven.

Matthew 5:19  Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.

Matthew 5:20  For I say unto you, That except your righteousness shall exceed the righteousness of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven.

Matthew 7:21  ¶Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.

Matthew 8:11  And I say unto you, That many shall come from the east and west, and shall sit down with Abraham, and Isaac, and Jacob, in the kingdom of heaven.

Matthew 10:7  And as ye go, preach, saying, The kingdom of heaven is at hand.

Matthew 11:11  Verily I say unto you, Among them that are born of women there hath not risen a greater than John the Baptist: notwithstanding he that is least in the kingdom of heaven is greater than he.

Matthew 11:12  And from the days of John the Baptist until now the kingdom of heaven suffereth violence, and the violent take it by force.

Matthew 13:24  ¶Another parable put he forth unto them, saying, The kingdom of heaven is likened unto a man which sowed good seed in his field:

Matthew 13:31  Another parable put he forth unto them, saying, The kingdom of heaven is like to a grain of mustard seed, which a man took, and sowed in his field:

Matthew 13:33  Another parable spake he unto them; The kingdom of heaven is like unto leaven, which a woman took, and hid in three measures of meal, till the whole was leavened.

Matthew 13:44  ¶Again, the kingdom of heaven is like unto treasure hid in a field; the which when a man hath found, he hideth, and for joy thereof goeth and selleth all that he hath, and buyeth that field.

Matthew 13:45  Again, the kingdom of heaven is like unto a merchant man, seeking goodly pearls:

Matthew 13:47  Again, the kingdom of heaven is like unto a net, that was cast into the sea, and gathered of every kind:

Matthew 13:52  Then said he unto them, Therefore every scribe which is instructed unto the kingdom of heaven is like unto a man that is an householder, which bringeth forth out of his treasure things new and old.

Matthew 16:19  And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.

Matthew 18:1  ¶At the same time came the disciples unto Jesus, saying, Who is the greatest in the kingdom of heaven?

Matthew 18:3  And said, Verily I say unto you, Except ye be converted, and become as little children, ye shall not enter into the kingdom of heaven.

Matthew 18:4  Whosoever therefore shall humble himself as this little child, the same is greatest in the kingdom of heaven.

Matthew 18:23  Therefore is the kingdom of heaven likened unto a certain king, which would take account of his servants.

Matthew 19:14  But Jesus said, Suffer little children, and forbid them not, to come unto me: for of such is the kingdom of heaven.

Matthew 19:23  ¶Then said Jesus unto his disciples, Verily I say unto you, That a rich man shall hardly enter into the kingdom of heaven.

Matthew 20:1  ¶For the kingdom of heaven is like unto a man that is an householder, which went out early in the morning to hire labourers into his vineyard.

Matthew 22:2  The kingdom of heaven is like unto a certain king, which made a marriage for his son,

Matthew 23:13  ¶But woe unto you, scribes and Pharisees, hypocrites! for ye shut up the kingdom of heaven against men: for ye neither go in yourselves, neither suffer ye them that are entering to go in.

Matthew 25:1  ¶Then shall the kingdom of heaven be likened unto ten virgins, which took their lamps, and went forth to meet the bridegroom.

Matthew 25:14  ¶For the kingdom of heaven is as a man travelling into a far country, who called his own servants, and delivered unto them his goods.

 

The Kingdom of God explained:

Just pulling out a few reference scriptures, to show that the Kingdom of God is also the state described above, as it starts in this earth, walking in the holy spirit of God, in fellowship with Jesus. . It continues into our eternal life.  69 instances of this term in the NT is to much to list.
 

Mark 12:32  And the scribe said unto him, Well, Master, thou hast said the truth: for there is one God; and there is none other but he:
33  And to love him with all the heart, and with all the understanding, and with all the soul, and with all the strength, and to love his neighbour as himself, is more than all whole burnt offerings and sacrifices.
34  And when Jesus saw that he answered discreetly, he said unto him, Thou art not far from the kingdom of God. And no man after that durst ask him any question.

Here Jesus is telling the Scribe he is close to the Kingdom of God, because he answered discreetly, and Jesus knew his spirit. . Hopefully this scribe did enter the Kingdom of God, after the cross began the 'time of grace', and the Holy spirit came to man.

Luke 8:1  ¶And it came to pass afterward, that he went throughout every city and village, preaching and shewing the glad tidings of the kingdom of God: and the twelve were with him,

Here Jesus is showing the new good news of the very near time of grace, as opposed to the time of Law, that the entire OT is built on, but as a shadow of the coming covenent of grace.

Luke 4:43  And he said unto them, I must preach the kingdom of God to other cities also: for therefore am I sent.

Luke 8:10  And he said, Unto you it is given to know the mysteries of the kingdom of God: but to others in parables; that seeing they might not see, and hearing they might not understand.

Here Jesus speaks to his 12 diciples, telling them that the Kingdom of God, will be theirs to know, but to others who cannot understand, He talked in parables. . as anyone under the law will never know the Kingdom of God under grace, but only as Jesus mentiones here is this verse:

Matthew 11:11  Verily I say unto you, Among them that are born of women there hath not risen a greater than John the Baptist: notwithstanding he that is least in the kingdom of heaven is greater than he.
 

Luke 16:16  The law and the prophets were until John: since that time the kingdom of God is preached, and every man presseth into it.

(He stated that we all press into this kingdom of God, and the word press means in the OT 'to lead down'
while in the NT 'to apply violence' which is what we sometimes have to do to get our body into subjection, by our will, with the help of God's will, not that we are actually violent with outselves, but it's a labour to deny our body at times, but the more we practice this, the more it becomes a labour of love.)
 

Luke 9:27  But I tell you of a truth, there be some standing here, which shall not taste of death, till they see the kingdom of God.

Here Jesus is saying that some who are still alive after he rises from death, then they will understand and enter the real Kingdom of God. preached by Jesus.
 

The Kingdom of Heaven, and the Kingdom of God, are literally interchangable in understanding.
This kingdom is spiritual in nature, and starts with the spiritual awareness of Christ, and the grace of His father God, and knowing he died for you. . it grows in us, according to our understanding, and faithfulness to God, through Jesus. . and in the next world it is us, the members of this kingdom, who will be reunited, and be forever with our Lord. . This kingdom is part of our inheritance. 

This was completely new to OT people, who built their whole existance on keeping laws and rituals,  given to them by the Jewish leaders in the synogoges.  Their whole justification was by how well they followed the law, customs, ordinances and commands of the church.  The only righteousness they could attain to, was self righteousness. . Now we are covered by the righteousness of Jesus, free, and with love.