Depending on how you read the achievement description, you might think this requires 9 enemies to kill. Fortunately, this is not the case - you just have to defeat 1 enemy with each weapon, so a total of 3.
You'll have to play for quite some time until you have obtained the 3 weapons. Now find an area with 3 small enemies. I'm not sure exactly when the timer starts, so I killed the first enemy with the ice mallet (as that was a little slower), then switched to the sword/shield and used the fire gloves last.
There are many places you can do this. I got it right after leaving Balmy Beach, crossing the chain and heading south towards the icy area. Right as you enter the cave leading to Peaky Peaks, there will be 2 slimes and 2 of the smaller worms that hide underground. I got 1 of the worms first as they can be annoying to target, then finished off both slimes. This is right near the transition point, so if it doesn't work you can go back to Balmy Beach and come back again - the enemies will respawn so you can try again.