All the melee weapons follow the same scheme: 3 blood metal, then 6 blood metal, then 7 blood metal AND a Blood of Valfaris for their final upgrade.
Fully upgrading any one melee weapon will require a total of 16 blood metal, and 1 Blood of Valfaris which doesn't appear in the game until partway through the 8th stage.
These are the applicable melee weapons:
Therion's Own (starting sword, medium speed, extra energy from hits)
The Bastard (heavy sword, shorter reach & slower swing, upgrades greatly increase dmg)
Blood Axe (faster swing speed, upgrades increase damage & add lifesteal*)
Jelly Whip (medium speed, damage increase with upgrades, longest reach by far)
Leg of Arach (intrinsic lifesteal, upgrades increase reach & dmg, reduced energy from hits)
Bathoryn (medium reach, high damage + dmg-over-time, late game/ng+ weapon)