Find something that's pretty straightforward, and not nonsensical (many songs are, especially what I've seen used in anime), and with an available translation. Install rikai-chan or the rikai version for whatever browser you use. Place the song in one window, and the translation in the other (if it's parallel; otherwise copy it into a word doc and make your own parallel translation). Listen to the song, figure out which sound is which to improve listening, and just follow the translation, then analyze the original sentence to see how it came to mean that. When you don't know a word, use rikai-chan. Maybe mine them afterwards . If you're more of a beginner/are learning a song which is mostly alien to you, spend a considerable amount of time only on that song and only move on to another when you've learned it well and could figure it out without the translation.

I'd also suggest finding a song which is more, erm, clearly spoken. Of course, lyrics are lyrics and any one will do, but if you want the full package find one where you can actually hear the singer speak the words, preferably at a pace which you can follow (ballads?)

I time the lyrics to the music using Aegisub and then run it through subs2srs. Takes a while to go through that process though. It would probably be easier to start with really vacuous pop songs and just look up the very few new words as you go. I was surprised when I first went to karaoke and everyone was singing these Japanese songs that I completely understood, despite not understanding any of the music I actually listen to at home.