Crawl

After you've played a couple times, you'll unlock the Deity Gor.

When you next start a game, choose "No One" when asked to choose your deity. You will then be asked which deity you want to taunt.

Taunting a deity puts you at a handicap. The type of handicap depends on the deity.

Gor's handicap is you start with the basic sling, and you're not allowed to buy any other weapons. You have to use the sling for the entire game, and have to beat the boss. You will NOT get the achievement if an AI challenges the boss three times and loses, even if you are in "first place" in the game over screen. You actually have to beat the boss.

Winning against the HARD AI with the sling may seem impossible at first, but it's not actually a bad weapon. Press the attack button once to prepare a rock and begin to aim. Then press again to sling the rock in the direction you're aiming. You can kite the AI monsters around, while attacking from afar. In fact, the rock you sling will often bounce off of one monster into another monster.

The thing to remember is: It starts off weak, but the monsters in the beginning are weak, too, so it's ok. However, as you play, the monsters get stronger, and the various bosses have a ton of health. Your sling, though, will remain weak. So try and purchase STRENGTH potions in the shop.

EXPLOIT:
There are two artifacts you can buy that actually allow you to switch your weapon.
One that switches your weapon with another adventurer
The other switches all your equipment with another adventurer.

Both of these artifacts can appear randomly at the store. There is nothing you can do to increase their chance of appearing. So if you see either of them, and you hate the sling, feel free to buy one.

I'm pretty sure this is not intended. The developer may patch this later, since switching your weapon defeats the purpose of taunting Gor. So if you want to use this exploit, you should act soon.

However, if this is patched OR if you just want to do it legit, the achievement is entirely possible without the exploit. You will become good with the sling the more you use it.

My successful run did not involve switching weapons. My shops didn't have the artifacts mentioned above. I'll post it here as an example of sling-gameplay. It is not the entire 20+ minute gameplay, just the last four mins or so, so you can see how I dealt with the boss, and as proof it can be done without the weapon-switching exploit.