Phantoms are a new enemy type that were introduced into the Onslaught DLC. They are similar in shape and size to hunters, but glow blue and turn invisible when they jump. To get this achievement, you will need to personally kill a total of 5 phantoms. There will be 1-2 phantom(s) that spawn in Area 1, and another 3-5 that spawn in Area 2. Another 10 or so will spawn in Area 3. This should for most people in a group of 4 to get the achievement in 1 complete playthrough of Nightfall.
You can kill phantoms in a variety of ways, much like any other enemy. Sentries/Vultures are effective, as well as traps. For this achievement it just comes down to progressing far into Area 3 and having many of them spawn.