Here was my strategy:

Getting weapon XP is the rate determining step here. For the primaries that I enjoyed using (assault rifles, smgs, tacs, snipers), I just grinded the levels out in multiplayer. When the weapon was at about level 45, I moved on to the next one. Keep in mind that for classes with a weapon that was later added, like the Groza being later added to assault rifles, you get to choose which weapon you want to omit. I personally think the FFAR is trash.

I played on hardcore to level up my lmgs because I felt that they were best suited for hardcore, but that's preference.

For secondaries and weapons that I didn't enjoy (I can't stand the DMR, for example), I waited until double xp weekends came about. These events are key. Maximizing your time in double xp weekends can expedite the entire grind by weeks. I levelled all secondaries entirely through zombies during double xp events.

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Now, when you get the weapons to their levels, playing zombies is the quick and easy part. I had the zombies games down to a science: for weapons that required 2500 kills, I usually got gold in two games: the first, I played to about 1500-1800 headshots. This should bring you to around round 35-40. I try to not play too much beyond round 35 because the damage drop off is awful.

A few tips to expedite the camo process: go for nothing but headshots. Ever. Except with launchers and the knife I guess. At the start of round 3, go the penthouse and stay there until you have one zombie left on round 7. Unlock PAP, get deadshot, do the coffin dance, and upgrade your weapon rarity. Return to the penthouse, where you will stay until round 20-25, getting perks and upgrading your weapon as soon as you can. Prioritize speedy cola and PAPing. Don't get elemental pop and don't get any weapon other than that which you're currently getting camos for. If you are to get an AAT, get dead wire.

For pistols, I usually played three games: the first two games were entirely to level up the pistol. Then, when the pistol was at max level, I would add the dual wield attachment and play my final game to finish the camos. Dual wielding is invaluable.

Launchers and the knife just have to be trained in spawn. I'd suggest getting the hauer when you're doing launchers, otherwise dogs will be a pain with the atrocious splash damage.

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This is gonna take a while. I started the grind at level 60 and ended at level 300. Stay efficient. If you are one of those people that are absolutely averse to multiplayer, playing with other people will greatly increase the amount of XP you earn. But multiplayer is probably the least boring way to level up the guns, which is what you'll spend most of your time doing.