Kingdom is God's rule and reign. As such, it is manifested in the life of the believer and, consequently, to the world at large. We want God to rule and reign on the earth just like He does in Heaven. We want his power and glory to be exhibited on earth just as it is exhibited in Heaven. We know that cannot fully happen (anymore than we can be perfect on this earth), but it is what we strive for nevertheless.
We are to "seek first His kingdom and His righteousness." Mt. 6:33 What is the object of our seeking? Church? Heaven? Nope. We are to seek His rule and reign in our lives.
Of course, every time a Catholic reads the words Kingdom in the Bible they think of the RCC. Cept those troublesome times that it doesn't really fit a church. Like when Jesus cast out demons and said that was evidence of the presence of the Kingdom (interested how y'all can twist that one to fit Kingdom=Church). Jesus also said that it is easier for a camel to go through the eye of a needle than for a rich man to enter the Kingdom of God. Now, I may be wrong, but I don't think there's many Catholic churches turning away millionaires. ;-) Finally, as Jesus was leaving Galilee for the last time, he sent 70 disciples to preach. He told them, "Heal the sick in it and say to them, 'The kingdom of God has come to you.'" Luke 10:9 Doesn't sound like the church to me :-)