***NEW FASTEST METHOD*** (CONFIRMED!)
There's a game braking glitch that will allow you to get this in next to no time:
Get 3-4 weapons a couple hundred XP away from level 5 and quit to Home. Go offline in the Network Settings and load up the game and get all those weapons to level 5 then quit the game. Delete the save from the console for Sunset Overdrive by clicking 'Manage Game' clicking the save game and choosing to delete 'From Console' ONLY! Go back online from the network settings, and load up the game and your previous save will be downloaded from the cloud and those 3-4 weapons will still be a couple hundred XP away from level 5 but you will have gotten progress towards this achievement in the achievement tracker. You only have to do this a couple of times to get the achievement.
This also works with the 50 gold challenges, and ALL of the collectible achievements!
but NOT for the 250 cloth items, as you have to own all of those at once.
(check respective solutions for more detailed information)
PS. My original solution was just a theory because i did this by accident, so this revised version is pretty much a copy paste of how D4 B1GD0G got this to work too, so a HUGE thanks to him for confirming this in the comments! :)