This is a really simple achievement that will most likely come while going for the other online achievements.
The requirement is to reach max level with a base game weapon. Each time you use a weapon you earn weapon xp which contributes to your level with that weapon. Keep using the same weapon, and you'll eventually get to the max level for that weapon.
There is no "best weapon" to use per se, but I found assault rifles and submachine guns easiest to use, but you may prefer marksman rifles, snipers, pistols, shotguns ect. Also worth emphasising that the weapon xp is specific to the weapon, not the weapon class. So xp earned for the STG won't level up the BAR for example.
Progress can be boosted with either weapon xp boosts, and I think weapon xp can also be earned in Zombies if you'd prefer, however I will verify this over the next few days. EDIT: yes, xp can be earned in zombies too.
For those of you that are unsure about which weapons count, this link will set out each base game weapon by weapon class, all of which should count towards this achievement. This is not my list and all credit goes to the people at GAMERTWEAK.
https://gamertweak.com/weapons-list-cod-vanguard/If there's anything I've missed let me know