Best Ways to Rank Up Fast In Modern Warfare 3

Play Ground War because there are more enemy players to kill. And this means getting killstreaks is generally easier than in standard team deathmatch. You also have a better chance of killing more enemy players with the AC130, attack helicopter, etc..
Use a prestige token (if you have one from other CODS) and unlock Double XP for two hours!
Play Kill Confirmed game type (as described above).
Use a variety of weapons to unlock weapon challenges. While you may have a weapon that you like to use all the time, try and use a variety of weapons to unlock challenges. Completing weapon challenges will ultimately make you rank up faster in Modern Warfare 3.
Use the Hardline perk in Modern Warfare 3 to get killstreaks easier because it reduces the amount of kills required to get a killstreak. Better yet, get Hardline Pro and 2 assists will count as a kill towards your next killstreak. Your deathstreaks also require one less death.

Specialist Strike Package to Rank Up Fast In Modern Warfare 3

If you are good at getting several kills in a row without the aid of killstreaks, use the specialist package. This is an excellent way to rank up if you can get several kills in a row because you get XP bonuses every time you get another perk. You get perks for killing several enemies in a row, but lose the perks when you die. With Hardline, you only need 7 kills to get all the perks.

This is definetly one of the best ways to rank up in Modern Warfare 3 IF you are able to get several kills in a row without dying consistently. Otherwise, it will not be beneficial to you.
Having trouble getting higher killsteaks in Modern Warfare 3?

If you find you are having trouble getting higher killstreaks like the AC130, set your killstreaks lower. You still get points for using lower level killstreaks. Note that killsteaks are set per individual custom class now. So you can basically have some classes that have higher killstreaks set and other classes that have lower killstreaks set.

Also remember that killstreaks stack in Modern Warfare 3. Whatever kills you get with a predator missile, attack helicopter, etc. will be counted towards the next killstreak.