Play hardpoint. Gets you tons of XP. Usually helps if you do well, but capturing a hardpoint is the easiest 200 xp you'll ever earn in your life.
EDIT: Due to a user suggestiing I go more in depth on how to gain xp and level up as quickly as possible, I'll do my best to get you to lvl 55 as smooth and fast as possible with the following tips:
-Contrary to what I said originally, FIND YOUR GAME MODE, whatever you like to play, have fun playing, or typically do well at, play it, as it will probably increase the rate at which you gain xp. The medals you earn after each game (like double kill, revenge, objective cap, etc) add to your xp total so every special action you do gets you more xp.
-For those new to COD games, each level requires progressively more xp to level up, meaning that it'll take a bit longer to level once you start reaching higher ranks.
-A big tip here is to use CHALLENGES to your advantage. The ones you will benefit most from by far are the weapon challenges. So, find a gun you like, use it a bunch, and as you level up the gun you unlock attachments and gain xp for completing the challenge to unlock the attachment. Once the weapon hits max level (level 20) PRESTIGE IT. You can do this by simply going on your create a class, selecting the weapon and pressing Y to prestige it. This will reset all your attachments and will allow you to earn the xp for unlocking attachments all over again. You can only prestige the same weapon twice.
-The last thing I could say is to put low scorestreaks like UAV, Hunter Killer, Hellstorm Missile, and Lightning Strike as they give boatloads of xp for being called in and getting kills with it. To get your scorestreaks quicker, I recommend using hardline and to PLAY THE OBJECTIVE. Black Ops II rewards the player so much for playing the objective and helping the team its insane. Now that they are no longer KILLstreaks everything you do has a certain amount of points attributed to it, and the amount of score earned for every action depends heavily on how important it is for that game type.
So, to wrap up:
Find your game mode, find your gun, prestige your weapons, get as many medals as possible, use low, team-assisting scorestreaks (like uav), and PLAY THE OBJECTIVE.
Hope this helps all of you get to lvl 55 that much quicker.