This can be done in any multiplayer game mode. I would recommend shorter rounds if you plan on playing for the 1000 games achievement. If you are just focusing on this one, then a 'lots of lives' and 5 minute match would be best.
The ship you select doesn't matter, but I chose the one with more energy. You must be sure to select the 'Quantum Slinger' as your weapon.
I would suggest using as many controllers as you can, as it gives you more targets to kill. Using four controllers and making it 1 (you) against 3 team game will also give your ship some additional firepower in the way of turrets that swarm around your ship.
As soon as the match starts, fly towards and enemy ship and shoot it a couple of times to decrease it's health. When it is reasonably low, hold the LT to activate your slinger and take out the enemy. Repeat this 250 times. Again, more players equals faster progress. In the challenge level, you are taught to use the weapon to sling enemies around. I found that in multiplayer, you can just hold the LT down and use it more like the Tesla Spike.
Also, I would recommend playing the medium difficulty of the Alien Swarm level. This will make a bunch of alien ships appear, which can easily be killed by holding down your slinger attack and killed by your turrets (if playing a 4 player local match). This will help towards your 4000 alien kills.