To get to Rank 20, you will need about 130,000XP. While this may sound daunting, it actually is not, and can be obtained in only one day of playing.
The fastest way by far (without boosting, but it is not really necessary for this) is to play Crash Site/Spears and GO FOR THE OBJECTIVES. In Crash Site, for every 5 seconds that you stay in the area of the pod, you will get 50XP, so crowd the objectives, and you can get at least 1000XP just from that. Plus all of your skill kills, your challenges, regular kills, assists, etc will all add up at the end of the game. At the end of a good game, you can rack up about 6,500XP.
In Spears, this is where the awesome XP is. Everytime you capture a Spear (it doesn't matter if you're first there or not), you will get 250XP. Just run from spear to spear and keep on capturing/killing those enemies. At the end of a good game in spears you can get around 8,000XP although my best was 12,000XP.
If you want to boost, it is the same method as in Crysis 2. Fill a lobby with 12 people in Capture the Relay. Have the entire team (except for one person on each team) stand around their own relay point. The one person from each team will go to the enemy spear and pick it up and drop it into the proximity of the enemy team so it gets reset. If you really need to boost this, then that will be your method.
NOTE: If you are going for all of the multiplayer achievements, level 20 will definitely come along the way. Completing the 25 challenges will take a while, and will score you some great XP bonuses along the way!