I’m not sure if this is 15 rounds at the start of the game, but coming from similar Black Ops 4 achievements, I did the first 15 to be safe.
I started with the proximity mine equipment, and a XM4, and didn’t really have much trouble until I got to the Megaton (he spawned on 15 for me). He isn’t too much trouble if you have a combat bow though.
I ran a couple attempts with a friend, and we died on 12 every time. The first game I ran solo I unlocked it at the end of 15.
If someone can verify that this is just 15 consecutive rounds, and not the first 15, I’ll update this solution!
EDIT : As Zenész has stated in the comments, this has to be 15 consecutive rounds.
As xCANExINSANEx93 has stated in the comments, it has to be the first 15 rounds, even though it doesn’t state it specifically.