These achievements require you to level your character to 20, 50, and 100. Leveling is achieved by XP which is gained from killing enemies. You should also keep as eye out for XP pick-ups which are occasionally dropped by more powerful enemies. These provide boosts to your XP. This will most likely be the last achievement you unlock as all others should be easily attainable on route to this one.
The fastest way to boost levels in this game is to play against hordes of Zombies on Bokor, whether you are racing for 100 or just need a little extra boost to beat a Revolution. The video below, found by VV1pCr3am, shows a great spot to camp with the Corona Ejector, at the 3:15 mark of the video added to the "Annihilator" achievement.