According to Catholicism what does it mean for every relationship in heaven to be perfect and fulfilling? The Catholic church teaches that every relationship in heaven will be perfect and fulfilling (for example, see this American Catholic reflection written by Friar Jim Van Vurst, O.F.M.).

What does this mean?

Will every relationship in heaven be perfect and fulfilling in the sense that


I will receive everything I want to receive from every relationship,
I will give everything I want to give in every relationship,


AND


everything that I receive and give in the relationship will be at the level I desire in heaven (everything such as affection, care, intimacy, etc)?
This is a huge topic, and I think, hinges on one of the central concepts of Christianity. I'll try my best to give an appropriate answer, but I probably won't be able to answer it fully.

I believe it's a mix of all three. The writer C.S. Lewis, although Anglican, wrote a fairly short fiction novel called The Great Divorce which exemplifies what many Christians (Catholic or otherwise) believe about the nature of relationship in Heaven. The main point goes something like this:


  "Heaven is reality itself. All that is fully real is Heavenly. For all that can be shaken will be shaken and only the unshakeable remains. - C.S. Lewis, the Great Divorce


Essentially, our earthly relationships are filled with corruptions of God's perfect love, manifesting themselves, for example, as lust, insecurity, fear, etc. Our need for relationship will never be satisfied on Earth because sin distorts our desires, but our earthly desires are a shadow of a deeper desire that can only be fulfilled in perfect union with God. 1 Corinthians 13 of perfect love. Verses 9-10 and 12 are also good examples of this "shadow of a deeper desire" concept.

Jesus will bring us into this perfect relationship. This idea is a central theme in Scripture in the form of the "Jesus will marry his Church" metaphor (see e.g. Romans 7, 2 Corinthians 11:2-4 , Ephesians 5:29-33, Revelations 19:7).

The Catholic church (and pretty much all "orthodox" denominations) believes that in Heaven - since we will have been sanctified - we will be able to experience real love, real joy, and real relationships without the corruption of sin. 

I highly recommend you read The Great Divorce if you haven't already.

Another way of looking at "fulfilling" or "satisfying" is in the fact that we can never have enough food or drink. We will always become hungry and thirsty again. Jesus addresses this fact to the Samaritan woman at the well in John 4:10-14, offering her "living water" that will satisfy her thirst forever, and to his disciples in John 6:35. Applying this to relationships is easier if you believe that humans have a constant desire for intimacy, love, affection, and acceptance, and if you understand that under the catholic view, that desire can't be fully satisfied by any human.