"What does it mean to have Christ in your heart? What does that feel like to you?"
To answer the first question, I would turn to Sacred Scripture; in the Epistle to the Romans, Paul exhorts us to put on Christ. This is sometimes translated as clothing oneself in Christ, and this is an important point to understand. To have Christ in our hearts is not just an internal abstraction, but rather should be obvious to the other people who see us, just as a garment is obvious to the people who see us.
To have Christ in one's heart is indeed a subjective state; we as individuals cannot fully express the internal dimension of our relationship with Christ when we have made room in our hearts for His divine love. And yet it is not solely a subjective state. There is an objective transformation of we who are subjects to the King of Heaven which we experience as subjects, and this is also apparent to those who come into contact with us each day. Having Christ in one's heart does not merely make us subjectively feel better; it makes us better in every important objective way and then we feel better as a result.
Christ's love fills our heart as we move other things out of it that are not of Christ. When that process is occurring, we will see in our lives that our old habits of serving ourselves go away, and new habits of ministering to others take their place. When that happens, we will see our habit of anger and self-righteousness replaced by a habit of mercy and compassion for those who hurt us. This profound transformation is often gradual, but it is noticeable by those who see us over the course of time. And perhaps more importantly, God notices it and delights in our closeness to Him.
As this happens, we will feel greater joy and peace. It's important to note that joy and peace are a product of our walk with Christ, not a precursor to it. And it won't happen in an instant when we get the formula right, because relationships are not formulaic. While it's true that good habits of showing love to one another, the routine exercise of mercy and compassion, may seem formulaic, its good fruits are varied. No loving relationship looks exactly like another in every detail.
I remember giving my heart to Christ when I was 8 years old, but merely doing as Paul says in Romans, confessing with my mouth that Jesus Christ is Lord and believing it in my heart, did not magically turn me into a saintly person in that instant. It did not make me a clone of Jesus Christ and save me from my sins by doing so because I said the right words.
Instead, it began a process of transformation, a grace given to me in order that I might set aside the fleshly lusts which are so often our outward garments and put on as clothing the divine love and mercy of Christ who sacrificed all so that we all might have the chance to live fully.
Public Domain, https://commons.wikimedia.org/w/index.php?curid=8921429
Note: The above is a painting of the Divine Mercy of Jesus, an apparition that reportedly occurred with St. Faustina Kowalska as a witness.