"But I'm a good person..."
When people bring up the subject of religion (particularly Christianity) and self-righteousness as in, "You Christians are so self-righteous," or something along those lines, the first thing to do is simply acknowledge the issue and say, "Yes, that's unfortunate. That's why it's so important for Christians to understand the gospel more deeply, because the gospel destroys self-righteousness."
Now you might have their interest, because the obvious question is, "How does the gospel destroy self-righteousness?"
Then you can explain: "Self-righteousness at its core is when any person says, 'My self is in good condition. My soul is good. I'm basically a good person. I don't need to be saved from myself or anything else.' In other words, 'My self is alright.' That's self-righteousness, a trust in one's self as being good enough for God and anyone else who happens to care."
The gospel teaches exactly the opposite:
yet we know that a person is not justified by works of the law but through faith in Jesus Christ, so we also have believed in Christ Jesus, in order to be justified by faith in Christ and not by works of the law, because by works of the law no one will be justified.
When the Apostle Paul writes, "No one will be justified by works of the law," he means no one is going to be accepted by God on the basis of their own moral goodness, their own self-righteousness.
That's humbling and it's meant to be. None of us are as good as we think we are. As a matter of fact, we're probably worse than we think. We have failed morally and we continue to fail morally. Your self is not righteousness enough to waltz into God's presence.
That's where the gospel comes in. We are justified (declared not guilty) by faith in Christ. We have faith in His work on our behalf. He lived the life we couldn't live. He died for our sins taking the penalty we deserve in our place. He rose again from the dead that we might live a new way -- by faith in Him, not by trust in our own self-sufficiency.
That's good news to everyone who abhors self-righteousness.