IMPORTANT NOTE: If you're going for the platinum, this is almost certainly going to be your last trophy, so don't go for this until you're done with every other trophy.
There are many different ways to attack the long grind for this trophy. You could go the much longer (but also probably more fun) route of playing multiplayer until you hit your 1000 kills. Botzone would also work in that regard. You could decide to do multiple full playthroughs until you have all of your kills. You could stick to one single contract and go through it repeatedly. It's truly up to you.
However, the most efficient way is to run through the 4th contract, Diplomatic Incident, over and over. With this contract, make sure you let Boris die when you reach the section with him, and try to kill every enemy while still going quickly. You should do this on recruit since it will be the quickest and the same amount of enemies spawn. The 5th contract, The Package, is also a good choice but not quite as efficient. Personally, I ran through the two of them back to back instead of just sticking to one to avoid getting burned out as fast.
You can check your kill count in the main menus if you want to.