What should I do to catch up on lore when I hit 90?

Patch 5.2 is the return of the Thunder King, Lei Shen.
Upon hitting level 90, head to your capital city and you should receive a quest to head to the Isle of Thunder. Completing this will open up a number of linear single player scenarios. Originally, these were spaced out by a number of days, depending on how long your server took to unlock different parts of the island, but this is no longer the case and you will play each scenario one after another.

These scenarios will help explain some of the lore surrounding the Thunder King. It will also unlock the island daily quest hub, which will also let you uncover some of the secrets! Once you have the required ilvl (480), you can visit the Throne of Thunder and fight through 4 LFR wings eventually encountering Lei Shen himself.

Patch 5.3 focuses on the escalating tensions between Garrosh and Vol'jin's Darkspear Rebellion.
Visit Lorewalker Cho at the Seat of Knowledge in the Vale of Eternal Blossoms to begin this story. This again will set you on a linear path explaining the ongoing problems between Garrosh and Vol'jin (you might remember it starting in patch 5.1).

Four new scenarios were also released in 5.3 which will help explain some things. They can be queued for using Looking for Scenario.

  • Battle on the High Seas
  • Blood in the Snow
  • Dark Heart of Pandaria
  • The Secrets of Ragefire

Wrathion and the Legendary Quest.
Be sure to also visit Wrathion at the Tavern in the Mists to pick up the legendary quest. Although you will be on the first step, it is well worth progressing not only for the story, but the legendary cloak and other goodies that come with it.

Blizzard have done a great job of making the story flow and at no point did I get stuck and not know where to go next. You might find spoilers in the links I included, so beware if you want to uncover these yourself.

Enjoy!