To get Persecutor status, you must kill a person 5 times before they have a chance to kill you once. Other people killing you shouldn't interfere with this. This involves about equal parts skill and luck. To get this achievement, you must manage that twice in a match, but not necessarily against the same person.
First off, try to get really good with at least 1 of the species. The Marines are going to be the easiest to handle for most people. Any chance you have to actually choose your race (Such as standard DM or mixed team DM), make sure and pick it. Change your race by hitting up until the picture of your character is highlighted then hitting A.
Usually there will be at least 1 person who chooses a race they are terrible with, sometimes there will end up being 2. These matches will end up being your best bet. Often times these players will rarely get kills themselves and become persecuted by several other players.
If you noticed that a particular player has a habit of visiting a particular area often, stalking that player or setting up an ambush in that area can be very effective. This tactic is easiest to pull off with the aliens or predators.
If you are a marine and can find a fairly open area that is frequented by other players (excluding other marines), this can make getting this achievement very easy. Just hide yourself near the edge of this area in a less traveled part of it and wait for melees to break out between the aliens and predators. When this happens, lay hellfire into all of them with any weapon you have. Often times you end up killing one, if not all, of the enemies involved in that fight. Any survivors will likely be too weak to take you on directly and will either run away or die in the attempt to kill you. On a side note, this is probably one of the easiest ways to get the Killer Instincts achievement as well, so long as you manage to get the right map for it.