To obtain this achievement you need to kill 10,000 zombies cumulatively across all matches of CoD Vanguard.
To quickly farm kills you can go to a Harvest objective at the Apartments area on Der Anfang and instead of filling up Runestones and completing the objective, just go to the Diabolical Damage perk fountain and watch between the entrances and just kill them as they come. The round will not change, the health of the zombies will stay at whatever round you are on, and you can infinitely stay here until you complete the objective. Zombies constantly drop ammo so ammo will not be a problem. The zombies will increase in speed and become more aggressive the longer you play so you will have to be on your toes the entire time, so if the area is too tight / narrow for your playstyle I recommend doing the Harvest objective in the area where the burning Windmill is at. It is also good to note that when I did this, I did not complete any of the three phases of the Runestones and I did not have any special zombies spawn in, but when I did complete a phase the special zombies started to spawn.
For setup, I recommend maxing Diabolical Damage at the least (constantly go for critical kills to conserve ammo and kill quicker) and possibly Health and Reload Speed. Use a triple pap'd weapon to also grind the 2,500 triple pap kills achievement as well (and for a further combination - use a weapon that you will use in Multiplayer to make progress on maxing a weapon's level quicker for that achievement as well!). I highly also recommend at least having level 2 armor and using the Frost Blast artifact. The earlier the round you grind the kills on, the easier this will be. I farmed kills on Round 11, but it can be done on an earlier round if you choose to as well. For covenants, I recommend using the ones that trigger slow/freeze on the enemy, more damage to slowed/stunned enemies, and increased damage on incremental shots on the same enemy.
This method took me about 6 hours to complete.