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