Best method to rank up fast: Play Well.
No, but on a serious note, i noticed that Hard point and Headquarters give me the most XP. I run UAV - Sentry Guns, and Guardian
Play Hardpoint with UAV/CUAV/VSAT. I get ~20000 a game without completing any challenges.
As others have said, Hardpoint with low Scorestreaks. Play with a good team, and know the rotations of the Hardpoints on every map. I highly recommend tac inserts as well. Also, cap the Hardpoint and get away from it, let the enemies cap it, then go in, kill them, and cap it again for extra points. Basically rinse and repeat and your SPM goes crazy high.
And yes, challenges are also important.