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