I just wanted to add a bit beyong Lita's method farming for kills. I can confirm that the DLC weapons count toward the achievement (I popped the achievement using Nothin' But the Hits).
If you're like me, you only have access to one DLC weapon, which puts the total weapon count to 22. The two I didn't do are Captain Ahab (it's extremely slow to level up) and the Charge Beam. DO NOT BUY IT. It costs 100,000 overdrive and all that will need to be saved to buy the collectible maps. I'm about halfway through the challenges and I can already tell you probably won't be able to buy all the collectible maps even after getting gold on all the challenges.
Overall it took me about 4 hours of farming kills. I had 3 of the weapons to level 5 by the time I was done with the campaign. If you try you could easily get more, but like Lita said, save this for when you won't be killing anymore.
EDIT: Let's do some math
Total mission OC ~305k
Challenges ~ 128k
Total = ~ 433k (not counting OC from trucks, kills, etc)
Maps = 225k
All weapons = 396k
Total = 621k
So that means if you wish you buy everything you possibly can, you need to earn an extra 188k OC from kills, trucks and Chaos Mode. That is A LOT to earn that way. So if you cut out the Charge Beam, you only need 88k. That is much more manageable.