Blackest Night is part three of a trilogy. Part one was Green Lantern Rebirth. Part two was the Sinestro Corps War. And then basically every issue of GL and GLC since then has been introducing the various Corps. I understand the most recent arcs on both titles were the most relevant (GL Agent Orange, GLC Emerald Eclipse). As for the larger DCU, you just need to know who's died in the past five years I guess.