To become attuned to Onyxia's Lair, you have to first start the quest "Dragonkin Menace" in the Burning Steppes. This quest leads through a chain that seems to end at "Abandoned Hope."

At this point, go back into Blackrock Depths and start killing Dark Iron dwarves until "A Crumpled Up Note" drops. The next quest requires drops from the bosses Golem Lord Argelmach and General Angerforge in Blackrock Depths. Taking the information back to Marshal Windsor leads to an escort through a portion of the instance (clearing beforehand is very beneficial). You then continue on to a confrontation with Onyxia in Stormwind. You don't have to fight the Onyxian Warders, but it can be fun.

To continue from here, you have to find Haleh in Winterspring. (There are notes on the page that thoroughly explain the path to her, at level 70, it's not a problem.) Haleh then gives you a quest to kill General Drakkisath in Upper Blackrock Spire.