There are exactly 1000 diamonds available in the game, but only three ways to get them. The first is exploration. You need to find all 221 diamond briefcases across the game. Each case contains one to three diamonds. Use the map here to keep track of the briefcases as you find them. The second is as payment for doing missions in the main storyline. The reward for each mission ranges from fifteen to forty diamonds. These cannot be missed as long as you finish the story. The third way is as a reward for carrying out the hit missions from the cell towers. These are always fifteen diamonds.
The cell tower missions are the ones that you need to be careful with. They are always missions to track down a target and kill them. If you don't kill them in the first attack, however, they may simply escape the map altogether. You still get a "Mission Complete" message, but you won't actually get paid, making it impossible to obtain all 1000 diamonds. To avoid this problem, just make sure that you have a save file right before you start the mission and if you don't see your diamond count go up as soon as you complete the mission, reload, and do it again.
If you find all the briefcases and do all the cell tower missions, you'll unlock this achievement before you get to the end of the main storyline.