As of writing this solution in 2018. The multiplayer is dead. If you are wanting to boost this achievement as well as several others, you will need to have the max number of players to get a lobby started. For all modes other than 2v2 Gnashers, you will need a full lobby of 8 players. 4 per team. This version does not support more than one controller. There is no split-screen.
If you are looking at completing this game I would save this and Level 100 for last. The best solution is 2v2 Gnashers boosting with 3 other players. You can idle this as long as need be, provided at least one person is going for their kills. I suggest going on a rotation to be fair with other players and to give breaks to others. The round time is only a minute so you should not be getting idle kicked either. You will only be getting 2 kills a round.
Seems to provide 150-200 kills per hour. So a very long grind to get this done with 3 other people. But it is doable and that's what matters. You should unlock your Level 100 along the way towards your 10k kills. You will have gotten quite a decent chunk of XP from boosting all the other online achievements.
If you have any further questions or concerns, feel free to comment below or send me a PM.