1. Speedrun the dragon as quickly as possible (helloyyy has done it in a blisteringly fast 1 minute 44 seconds)
2. Speedrun a perfect game/full board clear (i.e. only level up or grab a heart when you completely run out of hearts, so that you get as high a score as possible) (My best time in version 1.0 was 3:20, haven't tried in new version).
3. Beat the game with any of the 4 achievement stamps (click "next page" on the instruction booklet for an achievement breakdown).
4. Beat the game with 3 of the 4 stamps in a single run.
5. High score or speedrun with 3 of the 4 stamps.
6. Beat the all time high score with 3 of the 4 stamps (judging from the comments, it's _319(!!!!)_ points, without ever touching a rat, egg, or the lovers. But apparently 325 would be the all-time high score for the whole community).