Here is the full list of all 35 Backpack Hero achievements.
Carry at least 4 cursed items
Win a Quick Run as CR-8
Win a Quick Run as Purse
Win a Quick Run
Win a Quick Run as Satchel
Discover the Bramble
Carry at least 5 structures
Discover the Deep Caves
Find Prada's Locket
Defeat Disorder
Have at least 5 status effects at the same time
Discover at least 700 items
Carry at least 5 arrows
Unlock CR-8
Unlock Pochette
Discover the Frozen Heart
Discover the Magma Core
Unlock Tote
Stack at least 50 poison
Discover at least 400 items
Discover the Enchanted Swamp
Survive the Raid
Carry at least 5 manastones
Carry at least 7 consumables
Unlock Satchel
Wear a shoehat on your head and one on your feet at the same time
Unlock the Town Hall
Give a Tasty Fly to a merchant
Carry a helmet, clothing, two gloves, and footwear
Carry at least 5 shivs
Discover at least 200 items
Attract at least 40 residents to Haversack Hill
Win a Quick Run as Tote
Win a Quick Run as Pochette
Win a Run in Hardmode