A basic search on the biblehub indicates that this exact expression is only used in the NT, but the concept appears to exist in the OT (and, interestingly, it seems fairly Danielic).
http://biblehub.net/searchot.php?q=kingdom+of+heaven
And I'm surprised to see that the expression in the NT seems to only be in Matthew.
http://biblehub.net/searchnt.php?q=kingdom+of+heaven
Other NT writings use the same concept but call it the "Kingdom of God."
http://biblehub.net/searchnt.php?q=kingdom+of+god
So what is it? If I were to guess, I would say, based on the OT, that it refers to a time when the Torah would be universally observed on earth, including sacrifices, like in Zech. 14:16-21:
But does it mean something different in the NT/Christianity? My understanding is that Jesus was pro-Torah and anti-oral Torah, which is why he says in Mt. 5:20, "For I tell you that unless your righteousness surpasses that of the Pharisees and the teachers of the law, you will certainly not enter the kingdom of heaven," and in 5:23-24, "Therefore, if you are offering your gift at the altar and there remember that your brother or sister has something against you, leave your gift there in front of the altar. First go and be reconciled to them; then come and offer your gift."Then the survivors from all the nations that have attacked Jerusalem will go up year after year to worship the King, the Lord Almighty, and to celebrate the Festival of Tabernacles. If any of the peoples of the earth do not go up to Jerusalem to worship the King, the Lord Almighty, they will have no rain. If the Egyptian people do not go up and take part, they will have no rain. The Lord will bring on them the plague he inflicts on the nations that do not go up to celebrate the Festival of Tabernacles. This will be the punishment of Egypt and the punishment of all the nations that do not go up to celebrate the Festival of Tabernacles.
On that day holy to the Lord will be inscribed on the bells of the horses, and the cooking pots in the Lord’s house will be like the sacred bowls in front of the altar. Every pot in Jerusalem and Judah will be holy to the Lord Almighty, and all who come to sacrifice will take some of the pots and cook in them. And on that day there will no longer be a Canaanite in the house of the Lord Almighty.
What other meaning does the Kingdom of Heaven/God have in Christianity if not as a time when the Torah, including sacrifices, would be observed on earth?