I fail to see how Psalm 15 applies to Christians today.

What applied in the Old Testament doesn't apply to Christians today.

God is omnipresent so everyone already lives in the presence of God. Plus, if you're born again then the Holy Spirit lives inside you.

You can draw closer to God by engaging in spiritual discipline. You can pray everyday, read your Bible everyday, go to church regularly, take obedience seriously, listen to scripture based Christian music, and Christian podcasts.