First off, there are some easy ways to get some of this backstory yourself. Not only did the Jade Forest quests give you a good understanding of the state of the expansion, but patch 5.1's Dominance Offensive
and/or Operation Shieldwall
faction quests did an amazing job of unfolding each side's lore without being at all difficult to unlock new story elements. In addition, whether you're Horde or Alliance, you can run the 5.1 scenarios A Little Patience and Dagger in the Dark to see the events that helped set up the current animosity between Alliance and Horde as well as divided the Horde's loyalties between Garrosh and Vol'jin. Also, the Isle of Thunder daily hubs culminate in a conflict between Jaina Proudmoore and Lor'themar Theron which you'll definitely want to see for implications for patch 5.3 and beyond.
But wait, there's more! Just a few posts we've done that help round up the lore going into patch 5.3 for you (please note, there have spoilers for the story if you'd rather experience it in game):
These posts should provide a round up and overview of the lore of what's been going on all expansion, leading up to the events of patch 5.3 and explaining how and why we have Alliance spies and Horde agitators both moving to take steps to end Garrosh Hellscream's tenure as Warchief of the Horde. So now you're all caught up, and can go into patch 5.3 as either a proud Alliance agent, or a dedicated Hordeling out to rescue your people from a rogue Warchief.