XDefiant

Weapon XP Basics

Each primary weapon level requires 1,500 XP. (Was 3,500 pre-patch)
Without a booster, you get roughly 100 XP per kill, meaning it takes around 15 kills to level up a weapon.
Boosters double your XP gain, reducing the kills needed per level to approximately 8.
Assists also count towards weapon XP
You can stack a 2x XP token during a 2x XP weekend

Pick a weapon you're good with. AK and MDR seem to be the best Assault Rifles, Vector/MP7 the best SMG, and M249 the best LMG.

Additional Tips

Killstreaks of 3 seem to award bonus XP, but it's unclear if this directly translates to Weapon XP.
Avoid Escort Mode: Longer respawn times can hinder your kill rate.
Don't use an optic attachment; use the normal iron sights instead, and use the attachment slot for something that improves either recoil or ADS speed, depending on your play style.

Breakdown for clarity
Kills per level (no booster): 15
Kills per level (with booster): 8
Total Kills needed if you never use a booster (not taking into consideration XP earned by assists): 750

Source of the weapon xp numbers

You are awarded some boosters when you start the game and you can get an additional one on Ubisoft connect. There is also a free booster when you get to battlepass level 25.

Edit: There was a patch that changed the XP level needed to level up a weapon, as said in MoG Alucard212's comment "Level primaries has been changed to 3,000xp and secondaries to 1,000xp"

Edit 2: There seemed to be another update today reducing the XP needed even more.

Go on Ubisoft connect and purchase Weapon XP Booster for 50 coins / should be free if you have already a account

Weapon XP booster like I said in comments gives the player same as written a certain amount of time to get more xp out of any weapon so whenever the player is sweating have it on get the most out of it. it is only for the grind to go faster then natural progress depends on the player skill level of how much they can get out of it but its what I did and levelled up my gun by 22 levels in 2 hours and 20 minutes. YOU CAN ACTIVATE IT AT ANY TIME YOU WANT ITS TO MAKE THE PROGRESS/LEVEL UP WEAPON GO FASTER
If you want to go without this shortcut its the last achievement you will pop have fun playing with the same gun for 20+ hours

The best mode for this is Occupy, where all enemies converge in one point and you can get a lot of kills. Important to know is that, unlike in COD, you don't get weapon XP when you get regular XP, for example when capturing/holding a zone. Only kills and assists will give you weapon XP.

There's free weapon XP boosters available at the start of the game which you activate in the lobby. And there should also be at least one 1 hour double weapon XP available from a free challenge on Ubisoft Connect, but these challenges sometimes don't track properly. Finally there's also one available from watching eligible Twitch streamers, and there's permanent double weapon XP from Friday May 24th 10AM PT/ 1PM ET through Tuesday May 28th 10AM PT/ 1PM ET.