I've done this method 3 times now in the last few hours, it worked all 3 times. When I would look at my progress on the challenge and see it at 0, I just turned my controller off and browsed the Internet for 15-20 minutes. When I came back, I would be back to whatever level of the challenge I was at before the last match ended.
For example, today I got one assassination in the first game I played. The challenge reset, and I walked away for a while. When I came back, the number had gone back up to 1/3. I got an assassination in the next game I played, but again only 1. The challenge went back down to 0, and I walked away. I came back a while later, it was back to 2/3. Then I got one more assassination.
Don't know the exact details, why it might do this, or if there's other quicker ways...but it works. Just thought I'd share this tip if anyone is having trouble.