UPDATE 1: Thanks to Zoot Marlowe for commenting, that duel wielding weapons does make this go faster. Only some types of weapons can be duel wielded.
UPDATE 2: Thanks to Zoot Marlowe for commenting that all progress made does carry over to NG+.
This is will come naturally if your sticking to a certain weapon type. There are a total of nine different weapon types which are...
Guns and Swords
You gain progress as you use the weapon to kill enemies.
You can check your current progress by..
push the start button.
Go to your character name
Push X 4 times and it'll bring you to your Weapon Proficiency
Simply use a certain type of weapon and get it to the 1000 mark which will unlock this achievement, but also note there is a separate achievement that requires you to do all nine of these to max.