Start the game with an Atlantean. Don't bother picking up any weapons, or using any items, until you've earned the achievement. If you need to break mushrooms, switch to your empty hand to push through (though note that this process takes two turns per mushroom cap crushed, so best not to do it with hydras too close by).
There is always a specific mix of hydras on the first level. Most are straightforward to deal with, but the one with 3 heads and 0 resistance can be tricky to figure out without breaking the achievement with a Potion of Knowledge. Depending on the number of heads and their resistance to Ash weapons, take the corresponding sequence of moves while adjacent:
1 head: wait 3 turns, kill
2 heads: wait 2 turns, kill
3 heads, +1 resistance: wait 2 turns, attack twice to kill
3 heads, +0 resistance: wait 3 turns; attack the hydra 3 times; wait 1 turn; kill
Note that the 3 x +0 sequence will wound you for 24 HP in total, which can leave you a bit weakened for the second level if you intend to continue the run!