To perform any of these kills, you need to kill enemies who are not directly shooting at you. A Flank kill is when you kill someone from the side, who your partner is currently engaging. A Surprise kill is when you kill someone who is un-aware of your presence. A Tag Team kill is when you and your partner both cause damage to an enemy at the same time, to kill them.
You will know when you have successfully performed any of these kills as they appear on screen, after you have killed an enemy in one of these ways. To unlock this achievement, you must kill 250 enemies in any of these manners. These will accumulate over time as you play through the campaign. This will likely unlock without you realizing it, sometime towards the end of your first playthrough.