Try to start with 7 lives. [You must start with 5 lives or less to earn most of these achievements.]

Beat the entire game starting from 3 lives and 1 heart in the options menu without continuing.

Get the enemy score multiplier up to maximum on dry land.

You got the Octopus Ink!

Defeated the Octopus boss without taking damage. [7 lives okay]

You got the Spider Venom!

Defeated the Spider boss without taking damage. [7 lives okay]

You got the Moldy Cheese!

Defeated the Mouse boss without taking damage. [7 lives okay]

You got the Dragons Fire!

Defeated the Dragon boss without taking damage. [7 lives okay]

You got the Normal Underwear!

Defeated the Washing Machine boss without taking damage. [7 lives okay]

You've obtained the Magicians Gloves and completed the scavenger hunt!

Defeated the Magician boss without taking damage. [7 lives okay]

Earn 100,000 points without farming lives.

Collect 195 $s in the Bathtub before going underwater.

Collect 70 $s in the Attic. [see comments]

Collect 350 $s in the Lab.

Collect 100 $s in the Castle.

Collect 70 $s in the Basement. [see comments]

Collect 100 $s in the Freezer. [see comments]

Have ten lives in reserve without farming them underwater.