No idea what you mean by weapon skill level, or whether you're referring to skill level or number of uses.
Skills level up by being used, you can see this through the "xx - III has levelled up to Level xx" text that pops up whenever this happens. The maximum skill level is 10 times of its rank, which is signified by the roman numerals (III = 3 in the example above, hence max skill level = 30) located after the dash in the skill name. When a skill is maxed and the
skill meets certain requirements, Momotarou will evolve the skill for you, increasing the rank of the skill by 1 (or 2, in the case of skills like Idaten's Grace). Skills on the same weapon are not linked to one another in any way other than being on the same weapon, and weapons do not evolve.
If you're asking about how many uses each skill level requires, I doubt anyone can fully explain. Skills gain "exp", or whatever you want to call it, when they successfully hit, buff, heal and so on (i.e. the weapon loses durability). The amount of "exp" required for a skill to level up increases a small amount each time it levels up, and increases a large amount every 10 levels.
Please ask questions unrelated to bugs and crashes
here.