I don't know exactly how character Attributes work with weapon accuracy, but I know for sure you have to increase your skill to become more efficient with that weapon.
The more you use certain types of weapons, you will become more accurate with them.

Also, when levelling up, I believe there's a Weapon skill tree or something like that.
It allows you to put skill points and manually increase your skill with a particular weapon.
- Blaster pistols, blaster rifles, polearms, one handed melee weapons, etc.

Using a weapon and killing enemies with a weapon will automatically increase your weapon skill level for that weapon category thus increasing your accuracy and damage.