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