To earn this achievement that's called "Weapons Master", you must find all weapons and use them at least once.
Basically, this is how it works. In each of the ten levels, you will find different weapons to use against the bad guys. All you have to do is pick it up and use it against the enemy, whether you hit or kill one foe or multiple foes.To pick up a weapon, you just stand over it and hit B and either Billy or Jimmy (whoever you play it) should crouch down and snatch it up from the ground before standing up to their full height. Then you press X and it should be a weapon attack on the screen.
Below is a list of all the weapons and where you can get them.
Baseball Bat - Dropped by Williams' (the standard enemy) from level 1+, later becomes a metal bat
Whip - Drop by Linda from level 1 on
Gold Key (May or may not count, required for an achievement either way)
Hair Pick - Frequently found the hair of Afro Williams, from level 1+. Also found in trashcans from time to time
Knife - Used by Williams'
Barrel - Several are found in level 2, and every few levels after
Explosive Barrel - Also found in level 2
Robot Arm - Found in the environment of factory levels, level 3+
Throwing Fan - Dropped by any Ichisumi from level 3 onward
Throwing Dagger - Dropped by Shun (ninja girls) level 5+
Boomerang - Found on the ground frequently in levels 5 and 6
Sickle - Found on the ground in levels 5 and 6
Shuriken - Found laying on the ground in level 6 (one is where Fuzzface spawns)
Grenade - Frequently used by the red Lindas in the last half of the game
Spike Ball - First seen in the level 6 Killacopter fight on the ground, seen on the ground several times after as well
Pipe - Found on the ground in level 7
Acid - Found on the ground in level 7
Pistol - Found on the ground, or used by Scientist Williams in level 7
Eye Club - Found on the ground in level 9
Zombie Head - Found on the ground in level 9
Scepter - Found on the ground in level 9
Sai - Find on the ground, and weapon racks, in level 10
2x4 - Found on the ground in a couple of spots in level 10
Sword - Found in destructible objects/weapon racks in level 10
As you find each weapon, pick it up and strike a nearby enemy to make it count. That's all you have to do.
To make this easier for yourselves, you DO NOT need to complete the level for the achievement. Using the video I found and have attached to my solution, Starting with level one, I found the weapons for that chapter, used them, exited the level, and moved on to the next level and did the same thing over again. Once you hit an enemy with the last weapon on the list above, the achievement should unlock right away.
All credit goes to dynastygamesguias for creating and publishing this video for everyone to see. This is, IMO, the best video guide to use for the "Weapons Master" achievement. This Youtube user, dynastygamesguias, also has an achievement guide for any other achievements you're missing from Double Dragon Neon, so please be sure to check them out!
So, for the last time, find all the weapons, pick them up, use them once, then exit the level and continue on until all weapons have been used.
Quick note: From what I understand, you only need to toss the grenade to make it count, but to be safe, use it against one of the Lindas, anyway. Second, in level 10, you'll find a 2X4 that may be hard to spot, since it blends in the ground very well, but you'll find one nearby a destructible weapons rack sometime later on in Level 10, so be sure to spot it!
Have fun and good luck!