There are only 10 Weapon Challenges in Dead Island 2, and for the most part you should get these naturally. To check your progress in said challenges, hit the  button and then go down to Challenges > Weapon Challenges.

Just make sure, while you're playing the campaign, that you mix up playstyles a little and you won't have to grind out much in the end for this one. So, try to get Pistol, Shotgun and Rifle kills (50 of each). And then, try to get 50 Maiming kills (using a machete or an axe, etc.), 50 Headhunter kills (using a golf club or  a knife, etc.), 50 Frenzy kills (like the Wolverine claws and the bo staff) and 50 Bulldozer kills (using baseball bats and hammers) and you'll be fine.

Finally, the one that I was stuck grinding for a bit at the end was getting 50 Curveball kills - just make sure you're constantly using these throughout. I was clearly not!

It's also worth mentioning, I did have Challenge counter tracking issues after a while on my main Slayer, as they stopped tracking altogether. However, because some stats carry across playthroughs, I was able to continue progress towards those challenges on a different Slayer. Worth bearing in mind if you have issues.