The Burning Crusader

For those who don't want to look at each faction individually, here are the corresponding dungeons for the factions that give rep from mob kills.

Thrallmar:
Run Hellfire Ramparts and Blood Furnace until they stop giving rep, which is about 5999/6000 Friendly although some mobs in Blood Furnace still continue to give rep. Then run Shattered Halls normal and all three dungeons on Heroic until exalted.

Cenarion Expedition:
Run Slave Pens until 5999/6000 friendly, some mobs in Underbog will continue granting rep well into honored. However you can switch to Steamvault regular and heroic version to get your rap to Exalted.

Lower City:
Run Auchenai Crypts and Sethekk Halls as well as Shadow Labyrinth. Shadow Labyrinth is by far the fastest way to gain rep, ~2000 rep per run compared to about 750 and 1250 with the other two. Heroic versions will give rep all the way until Exalted but Auchenai Crypts stop giving rep on normal at 5999/6000 friendly and Sethekk Halls goes up until end of Honored.

Keepers of Time:
Just farm heroic Black Morass by letting the final boss kill Medivh and repeating the entire 17 waves all over again. Otherwise you can run Normal and Heroic versions of Black Morass and Escape from Durnholde for rep.

The Sha'tar:
Run The Botanica, The Arcatraz and The Mechanar normal and heroic all the way until exalted.

The Burning Crusader would be a great title.

Reputation with these Factions comes from doing their respective dungeons (normal and Heroic, where Heroic grants more reputation), as well as from doing all the quests related to that Faction.

See the specific Faction pages for more detailed information about gaining the reputation needed for this achievement.