Heroes of Might and Magic VI (Or whatever anagram thereof they're calling it) tracks Weapon experience separately from Dynasty Experience.
Dynasty Levels go up to 50, and unlock Tiers at the altar of wishes.
Weapon Levels go up to 5, and unlock new passive bonuses or active combat abilities.
In both cases, however, leveling them up is the same - winning battles. In the case of the weapons, the dynasty weapon in question must be wielded by a hero in the battle.
So... what do you have to do? Win more battles! You've not reached Weapon level 2, you've reached Dynasty level 2.
Final Note: Don't be discouraged by the seemingly insane numbers you need to reach weapon levels 3, 4, and 5. Is it a lot? Yes. But in the last map of each campaign, when you're routinely slaying 100's of angels (or angel-analogue) per fight, it adds up quick.