Your goal in this is to rack up enough points to be awarded with the best title after entering your initials at game over. I achieved this with 19400 points, but with the title thresholds being 5000 points each, you might only need to get over 15000 points.
You gain points by letting hammers fall on your character's head. The cyan coloured ones give you 200 points and add to a pain meter (you can withstand up to 10) and the red coloured ones clear the pain completely, with a point bonus that scales with the amount of pain removed.
There's a bag of money at the right side of the screen, if you pick it up and take it to the left, it gives you 500 points and restores one of the shelter blocks.
At 10,000 points, you are awarded an extra life.
The controls in this game are very much tied to the draw rate of the screen, and take a bit of getting used to. Also, dashing under a hammer at the last moment makes it fatal. You have to let it fall on you. This and the very random rate of hammers make this achievement quite a challenge.
I'm not sure whether emulator save scumming works for this, it might do. I didn't try that.