This can be done online against human opponents or in a local Bot Match, though the former could prove difficult if you're not boosting as there is no player base to speak of. Even if you successfully find other random players, this achievement will be challenging since it's unlikely that they pass on purchasing items as well. With that said, boosting is recommended, though the below strategy applies regardless.
Once you've landed in your Drop Pod, head forward and await the countdown to start the round. Usually at this point you'd peruse the shop for perks, but we'll need to forego that for now to get this achievement. While you'll have no perks, you'll still have all of your abilities at your disposal. You can come back shortly to purchase them if you're planning to earn other achievements in this match as well.
Rush forward with your fellow Awesomenauts and assist in damaging the enemy opponents. Once their health is low, they may attempt to retreat but you can stop this from happening by using a crowd control ability such as my main Raelynn's Timerift, which slows the enemy and does minor damage. If they manage to get out of reach behind their teammates, Raelynn also has a useful Snipe ability for finishing off distant escaping enemies.
If you fail at first, chances are other enemies were weakened in this initial brawl, so try to land a killing blow on one of them. If you're unsuccessful early on in the match, it will only become more difficult since your opponents will be earning Solar and spending it in the shop. Again, boosting this is highly recommended. Good luck!