Blood Hunters only seem to form short-term "bonds" with their weapons
The description of the Blood Hunter class does indeed say:
They surrender their own vitality to form a bond with their weapon, allowing them to harness the elements in a whirlwind of dangerous strikes.
This is referring to their Crimson Rite class feature, where the Blood Hunter sacrifices some hit points in order to power up a weapon they're wielding. However, these "bonds" are quite temporary and not permanent, as the rite describes (emphasis mine):
As a bonus action, you imbue a single weapon with the elemental energy of a known rite until your next short or long rest.
Should your weapon leave your grip, the rite fades immediately.
So the bond they form only lasts for as long as they're holding on to the weapon, and even then only until the next time the character rests. It doesn't really represent a long term bond of the kind you're imagining. However there are other options in the game which could fit with this kind of concept.
Pact of the Blade Warlock
A class which does have that kind of association with a weapon is a Warlock choosing the Pact of the Blade Pact Boon. They have the ability to summon a "pact weapon" and can even perform a special ritual to bond themselves to a specific magical weapon, which might be closer to what you're thinking of! If you're a Critical Role fan, you might recognise this as being the class and option that Fjord has. A Warlock always has a mysterious patron who grants them their abilities and knowledge, but it could easily be the case for a Pact of the Blade Warlock that their patron communicates with them through their pact weapon.
Sentient Magical Weapon
Alternatively, it is possible for magic items to be sentient individuals in their own right, as described in the Sentient Magic Item rules. You could easily imagine an ancient magical sword with its own personality, acting as a mentor to the young adventurer who found it. It'd be very unusual for a low-level adventurer to have that kind of powerful magic item in most circumstances, but if your DM really likes the idea and thinks it could be interesting he might be willing to work something out with you - if not starting out, maybe planning for your character to find such a weapon as you play so you can explore that kind of story.