There is the obvious way: Mostly or only use one weapon to kill everything until it finally hits level 5, then repeat with 9 and 19 more weapons to grab the other two achievements that involve leveling weapons to 5.
There is another way though: Some weapons have a challenge associated with them. In these challenges, you essentially get infinite ammo for the weapon during the challenge, so you can save yourself from having to buy more ammo for those particular weapons. This method is tedious, and doesn't work for every weapon, but is the most cost effective for the weapons that it does work for.
I personally recommend a mix of grinding and just playing (mostly playing). If you put all of your favorite weapons on the weapon wheel and just use all of them as necessary (not all weapons are effective against all enemies), it shouldn't take incredibly long to level a weapon to level 5.
Bonus tips:
Once you level a weapon to level 2 and can put an Amp on it, make sure to put an EXP amp on it if you are only focused on leveling it.
I saw the "Captain Ahab" listed as a weapon that is hard to level. I actually love this weapon and it is VERY effective against herkers, spawners, and other large OD. Using it only against these enemies, I am close to getting this weapon to level 5 without searching for large enemies.
If you have a more effective way of leveling weapons, feel free to leave it in the comments and I will add it to my solution if you want me to or you can add your own solution to this achievement.