To obtain this achievement, you must ensure that you are using a Specialist Ability, as opposed to a Specialist Weapon like is required for the 'Deadly Specialist' achievement.
You have to get 5 kills while using a specialist ability in a single game. This can be done with multiple uses of the specialist ability in one game.
The best option that I found for getting this achievement is Nuketown Domination, if you have the Nuketown map. This is because 6v6 on a small map ensures that when you use a specialist ability, the odds of finding multiple enemies in the limited time frame are greater. It is entirely possible to do this on other maps if you do not have Nuketown.
Domination I believe is the longest gametype if one team isn't completely dominating the other team, so it is entirely possible to get 3-4 specialist abilities activated in a single game.
I highly recommend Specialist Battery - Kinetic Armor for this achievement, because when it is activated, you have approx. 7 seconds where you take no damage from body shots and can more easily avoid being killed while getting a few kills in those 7 seconds. But other specialist abilities, such as Ruin's Overdrive, work just as well.
Playing objectively (capping domination points), killing enemy players, and getting assists charges your specialist ability. For a small map like nuketown, I highly recommend focusing on kills rather than the objective point in the middle, as it usually results in death and wastes precious time that could have been spent killing someone.
If you vote negative, let me know why so that I can edit this solution if it isn't up to standard. This solution has worked great for several people I know already. (It does take some work)
Edit: epic fail 1710 pointed out that safeguard games generally last longer than domination matches, especially if there is a draw after the first 2 rounds. This makes it possible to use your specialist ability maybe 5+ times in one game. Thanks for the input!
Edit2: The Overclock Perk at level 20 reduces the time to get your specialist ability back to use again. Thanks for reminding me REDWINGS05