Call of Duty: Vanguard

Note: Even though this achievement description may imply it needs to be done in Multiplayer, be aware that all zombies actions (kills, challenges, etc.) contribute towards this level too.

Note 2: Even though Seasons and the Battle Pass reset every two months, your gun rank does NOT reset with it. You therefore don't have to worry about fulfilling the requirements for this achievement within that two-month period of a single Multiplayer Season.

Max level with a primary weapon is level 70 (max with a secondary weapon is only level 60, but I highly doubt you'll be leveling up a secondary weapon before a primary). Since this is such a high level, it's in your best interest to pick a primary weapon you like most/are best with, and stick with it as much as possible. On top of that, every operator you can play as has a favorite weapon. Pairing an operator with their favorite weapon gives you bonus operator XP (beneficial for Top Of The Mountain (35G)) as well as bonus gun XP for this achievement. So once you're decided on a gun and its corresponding operator, you'll then need to complete a challenge to unlock that operator. NOTE that every operator has one specific challenge in Multiplayer OR a different specific challenge in zombies. Pick whichever of the two seems easier or faster, since you'll want to start with your chosen operator and gun as early as possible, since both take a while to level up. For example, I chose the Sten SMG, which is Arthur's favorite gun. I unlocked Arthur by successfully exfiltrating three times in Zombies, so I could use Arthur and the Sten every time I play. Be aware that unlocking an operator doesn't automatically give you access to their favorite gun. If a gun is unlocked later, you'll need to rank up before you have access to it. I therefore recommend you choose one of the guns available from the start so you can start working on leveling it up immediately. 

With all of that done, you can finally start (efficiently) working on reaching max level with a gun. As noted above, Zombies mode contributes to your gun level. As a result, if you want to maximize efficiency (and minimize time spent in Multiplayer), it makes sense to first unlock all Zombies achievements with your chosen operator and gun, since all the time you spend in Zombies unlocking those achievements will unavoidably level you up a decent amount. Once that's done, it's up to your personal preference of which mode you prefer to finish leveling up: Multiplayer or Zombies. In my opinion, Zombies is far easier and faster to level up, but possibly far more boring. Either way, the fastest way to level up in either mode is to focus on completing as many gun challenges as possible, since they give you big chunks of bonus XP. Take a look through them (there are separate challenges for Multiplayer and Zombies)  and try to work on as many as possible. Since you can mow down Zombies without much skill, completing gun challenges in Zombies is, in my opinion, far faster than reaching max gun level via Multiplayer, but do whichever you prefer.

If you do all Zombies achievements first, depending on when you unlocked your operator (to get bonus gun XP), you may be as far as halfway done with this achievement just for finishing the Zombies achievements (I was level 38 with my Sten, and I didn't even unlock my operator as early as I could have). From there, focus on challenges in whichever mode you prefer, and be prepared to spend multiple hours in each mode, until you finally finish and reach level 70.