I've been trying all the video guides that can be found online without much success. They all seem to require lightning-fast reflexes that the somewhat poor controls in this game can't handle and I just end up frustrated, so I modified PaleVoyager's guide from YouTube to be slightly easier to pull off. The hard part of his video was the very first two kills and I found a way around that using a crab. This method is less elegant, you don't get to buy everything and it's not the fastest out there, but it will get you the achievement with little frustration.
Remember to dash through every fight.
01. Move one square up, one square right. Fight 1 jelly, enter first town.
02. Pray in town.
03. Walk straight up from town to a crab enemy, kill it and walk back to town.
04. Heal, pray and buy the Gladius.
05. Walk left, fighting 1 jelly immediately outside of town.
06. Walk left after fighting jelly, fight 1 snake and then dash to second town.
07. Pray, do not heal yet.
08. Exit second town, move 1-3 squares to the right to fight 1 snake and then retreat to town to pray. Heal when needed.
09. After 2-4 snakes killed, you will start killing them in 1 hit. When you do, keep walking right until you run into the second snake, then mad dash left to the second town.
10. Repeat until you have somewhere around 3000 gold and are level 13-15, then go to the second town and pray.
11. If you have around 2500 after praying, dash to the southern town and immediately enter.
12. Buy Hero Sword but don't pray.
13. Leave southern town and kill one beastman.
14. Dash back to town and pray.
15. Head directly left from the southern town and try to encounter two beastmen per attempt, then run back and pray.
16. Once you're around level 17 and have 2000-3000 gold or so, dash north to the fourth and final town. (Note: this may be just 1-2 kills after you reach the second town. You don't have to hang around the southern town for very long.)
17. Heal, pray and buy the armor in the fourth town. (You might need to skip the armor unless you level up in step 18.)
18. From here, you can either go straight to the boss fight or hang around and level up using the tougher monsters available here. (It is probably a good idea to level up once or twice here - I didn't and very barely beat the boss with no time left and almost no HP.)
18. Dash through the boss and win!
PaleVoyager's video is linked as a reference. All credit for it and the basic strategy go to him.