I am a bit obsessed with Cohen. I have been for many years. I remember the first time I was exposed to his music: it was at a coffeehouse, and a guitarist played a cover of "Hey, That’s No Way To Say Goodbye." I went out and bought Songs of Leonard Cohen, listened to it, and then went back and bought every album of his the store had. Cohen's music is so rich with allusions and poetic ambiguity, that I don’t think Cohen himself could give an exhaustive explanation. Rather, it has many nuances and is more thematic than it is a cohesive narrative, so I think it is better understood like a poem than a story.
If you aren’t familiar with them, the passages most obviously alluded to are David playing the lyre (1 Samuel 16), David and Bathsheba (2 Samuel 11), and Samson and Delilah (Judges 16), among others. In verse one ("I've heard there was a secret chord..."), I believe Cohen uses the symbolism of King David to refer to himself and his creative process, and he is addressing the listener directly: Cohen is the baffled King David. He is describing the structure of the song itself. In verse two ("Your faith was strong but you needed proof..."), I believe Cohen is now addressing himself, in a kind of introspective and self-critical monologue, in which he alludes directly to himself as David and then as Samson. Both are references to sensuality and brokenness in romantic relationships (one of Cohen's favorite themes), especially in the last line of the second verse.
The third and fourth verses ("Baby I've been here before..." and "There was a time when you let me know...") are focused on this theme of broken romance and the ambivalence of love. I believe the verses starting with "It's not a cry you can hear at night…" are referring to someone in love, and Cohen is dispensing the poetic notions of it, as he both praises and laments it, drawing parallels to a troubled faith. I love the fourth verse. Cohen directly confronts these themes, and provides a kind of definitive opinion: "There's a blaze of light in every word / it doesn't matter which you heard / The holy or the broken Hallelujah." The duality of the experience of love—suffering and ecstasy—are both the authentic experiences of the human condition—we are most human when we suffer as well as when we are blissful on account of love. The final verse, in my opinion, is again putting love back on a pedestal, although it is a more authentic love—it is praised despite the suffering it creates, so central is it to the human condition.
I feel like I've only scratched the surface, and a more literary mind could better interpret and express it than myself, but those are my thoughts. If you know Cohen, you know he is all about religious allusions and frequently embodying scriptural figures, as well as his ambivalent and intense views on romance and sensuality. It is truly a beautiful song, one of my favorites. (Just as a point of trivia, the song was met to a very cool reception, until Bob Dylan popularized it by performing it live, long before there were any covers.)