For me, this came naturally as I progressed through the game on my first playthrough (Hard). It just so happened to be the Constructo Blaster that was my first Lv.5 weapon, but as I began to unlock other weapons, it slowly lost its power. All weapons level up from mostly kills with them. Some odd exceptions are the Grooviton Glove and the Cryomine Glove, which require you to use the weapon and THEN kill them (probably conserving ammo by using your wrench).