I don't think anyone has mentioned this yet, so I'll throw this additional tip out there for what helped me get my weapons up quicker than any of the guides above.
Set whatever weapons you plan on upgrading, stick to let's say 3 at a time. Play Chaos Squad and select Downtown. Get to the Night Defense mission(and make sure you saved your ammo in the guns that you are trying to upgrade for this part). When it starts, run to the blockade where the enemies are starting off. They usually go to one blockade at a time. Spam your weapon into the horde of enemies and you will level up ridiculously fast. I was able to bring 1-3 weapons from a Lvl.1 to a Lvl 5 in each run through this night defense.
I think you should only do it on the Chaos Squad, I tried offline and it didn't seem as quick.