All you need to do is complete every level of the game, don't worry, there's no missables. As you progress you need to collect every Puzzle Piece and every Blue Stone.
No missing piece! Trophy - Obtain an special relic by finding every jigsaw puzzle piece in the caves!Puzzle Pieces - There's 5 in each level, most of them are visible, but some you need to use the some character ability to reach it. Try not to finish the level until you find each piece.
As good as Chester Copperpot! Trophy - Obtain an special relic by collecting every blue stone in the game!Blue Stone - They serve as life, so if you get hit, you'll lose them. The worst enemies are the Big Moles, as they can hit you more easily. If you get hit, just let yourself die and repeat the level.
With this, you need to beat every boss without getting hit too, the good part is that if you get hit in the final levels of areas (1 to 3), you just need to let yourself die and you'll start at the beginning of the boss fight with all the gems you collect before the fight. The worst part is this last level (4-4), since you'll need to beat every boss again and the final one, without getting hit.
This video here shows how I did that: