Leveling up in the game is based on earning XP, which you gain from completing contracts and killing/capturing enemies. Killing enemies with headshots earns you double XP for that kill. You can also earn a small amount of XP for completing subcontracts. Also, you should focus on completing the daily co-op challenges. They provide HUGE amounts of XP when you complete them, anywhere from 2,000 to 10,000 per challenge. These are an absolute must if you want to level up quick! The bulk of the XP will just come from completing contracts. This achievement will take the longest of any in the game.
Some tips: Use medkits as much as needed to complete missions. Buy a good rifle with a large clip to rapidly dispose of enemies (Anti-Material Rifle is best). Go for headshots for x2 XP. Make use of the cover button, especially when reloading. Shoot exploding barrels when possible (though you miss out on headshot XP). Some informants regularly give better missions (more XP/cash) then others...Make note of which give the best contracts. I find Shady Wessler gives the best so far (though he costs 5 credits to unlock and gives almost always a capture mission).