Phantoms are one of the new alien species in Extinction mode. The Phantoms are distinguishable by the blue glow in their eyes, their large size (about the size of a Leper), and their teleportation skill. The Phantom can take a lot of damage so I recommend the LSAT with explosive ammo. The LSAT can be found in the first area behind a hive on the second floor area near the catwalk. This is the same room where there is a locker and an SOS radio transmission.
The toughest part about fighting the Phantoms is that they will teleport away from danger if you fire at them or if you get too close to them. However, if you stand still, the Phantom will teleport to you to inflict melee damage. At this time, unload your LSAT with explosive ammo to kill him. The Phantom may teleport before you finish him, but usually the explosives will make him stumble which gives you enough time to kill him. Repeat this five times throughout the game to get this achievement.
Edit: I have done much more testing on this achievement and I can now confirm that the person on the team getting the achievement must deal the final blow to five phantoms. Each player in the game has their own 5 phantom kills to do for this achievement. If you have any more concerns or comments, please post them in the comment section before voting negative. Thanks to everyone that has commented so far for their contributions.