I managed to do this on a PVE server after Cata came out on my prot warrior. I'm a lousy PVPer and my character was killed by other players way more than 15 times. Doing this quest tested my patience but I'm a bit of a completist, so I stuck with it.
I initially picked the quest up at the time I was doing the rest of the quests in Icecrown, roughly in July 2010. However, I did not seriously pursue the quest until I hit level 85 in February 2011 (I do not level quickly). Here is how I did it.
First, I didn't abandon the quest.
My basic strategy was simple: hang out on the Alliance gunboat, the Skybreaker, and kill npcs until ally players attacked me. More often than not, no allies responded at all except possibly to watch. Even when they did, they often killed me, especially if they were fellow 85s. However, occasionally a level 80 ally (or less) would attack me, and usually I could kill them. (The fact that I couldn't always get to them is a testament to my incompetence in PVP and the lousiness of my gear for much of the time.) Today, a 77 dwarf priest attacked me and I was able to kill him four times (thus bringing the total to 15 and confirming that my character is a bully), so there does not appear to be a cap on kills any more; at least, if there is, the cap is at least four.
Most of the npcs on the Skybreaker did not attack me even when I stood next to them. In any case, there are no npcs on the Skybreaker that should pose a serious problem to a level 85. However, there are certain aspects of the fights on the boat that can cause problems if an ally player decides to attack you.
The skybreaker marines throw a net on you which immobilizes you for a few seconds. This is infuriating when fighting ranged classes. For this reason, take out the marines quickly.
Knight-Captain Drosche on the top deck does not hit hard, but has a fair number of hit points, leading to an incredibly boring fight. Ranged ally players will attack from the stairs and then run away; if you pursue, you may end up with more npcs than you can handle.
In particular, Thassarian, stationed to the left and below Drosche, does significant damage which can take you out if you're distracted by Drosche's tedious attacks or an irritating player. Indeed, Thassarian is the only fight on the boat that is remotely interesting. I suggest you fight him under the stairs in case a ranged ally attacks you from Drosche's deck.
14 of my 15 player kills were obtained in the above way. None of the players concerned was above level 80, but only one of those 14 kills was obtained by attacking a player who was already flagged; in short, they chose to attack me, for reasons unknown to me. (Irritation, stupidity, alcohol, honor...)
I made myself a tempting target by holding back while fighting Thassarian or Drosche. At least two of the kills were obtained in this way. Also, if at all possible, I made sure to kill them again when they come back for their corpse. (The more savvy ones resurrected far away from me and immediately mounted up. Those missed opportunities were most upsetting.) If they were afk and flagged, thinking perhaps that the Skybreaker is a safe haven, I gently remind them that this assumption is incorrect by one-shotting them.
In the unlikely event that an ally was attacking our gunboat, I flew over to our boat to defend it. That was how I obtained the remaining kill and was the only time I killed another 85 in this whole sorry saga.
I've done it once; never again... my poor warrior spent most nights on a flying mount in Icecrown rather than a nice bed in Dalaran. I did experiment with other places in Icecrown, but the gunships were really the only places that seemed to work for me.