Takes a Licking (30)
Bashed 30 Tickers
Tickers are the small, tick looking explosive beasties. To bash one, equip something other than the Lancer or a grenade and perform a melee attack on them (B button). You'll know that you bashed one when it goes flying in the air. Tickers appear throughout the game and in Horde mode, so if this one doesn't pop after doing both the campaign and 50 levels of Horde, just load up Act 1, Chapter 5 on Casual and get to bashin'.
For all of the following weapon achievements, you can get them in any mode, be it multiplayer, co-op campaign, single player campaign or Horde mode. They're also cumulative across the different modes so you can get as much done in the campaign or Horde mode and then play clean up.
Organ Grinder (10)
Killed 30 enemies with your Mulcher turret
For this one to count, you have to mount your Mulcher, the chain gun carried by the Grinders, and kill enemies in any mode. To mount your Mulcher, get to cover and press the left trigger. Your character will mount the Mulcher on the cover and you can fire away. If you peek out from the side of cover with the Mulcher, it ain't mounted. Again, Mulcher opportunities abound in campaign and Horde mode, however if you don't get it by the time you've finished both you can use the split screen multiplayer to your advantage. Get another controller and sign in with another profile. Start up a local multiplayer match on Gridlock and use the weapons substitutions to replace the Longshot with the Mulcher. Pick up the Mulcher and either hoof it to your opponent, or use the other controller to maneuver them close to you. Either way, mount the Mulcher and kill them. If you don't want go to the second controller route, start a match with bots and kill them. Obviously, you can use any map you want, but Gridlock is small, symmetrical and the weapons are easy to find. To make it even easier, have someone else take up the second controller and watch those kills fly by!
Shock and Awe (10)
Killed 30 enemies raining down death from the sky
For this one, kill 30 enemies with the Heavy Mortar. There are a bunch of opportunities to do this in the campaign, particularly in chapter 4, however you probably won't get all of them there. Get as many as you can in the game, and then either load up the mortar heavy parts of the campaign, or follow the same strategy for "Organ Grinder".
Said the Spider to the Fly (10)
Effectively used grenade mines
This one is kind of tough to get in the campaign. A grenade mine is a grenade that has been stuck to a surface by equipping it and then performing a melee (B button) attack next to said surface. Once planted, any enemy that walks next to it will get a face full of boom-boom. Seeing how the enemies in the campaign aren't real big on the bum rush, you'll probably have to resort to the strategy for "Organ Grinder" for this one. Use the weapon substitutions to load the map up on frag grenades, plant one next to your enemy and then use the second controller to walk them into your explosive trap. Take out ten enemies with your grenade mines to net this achievement.
Crowd Control (10)
Got up close and personal with the Boomshield
The Boomshield is the shield carried by the big mace wielding dudes in the campaign. Once they're dead, you can take their shield and use, it, however melee is a dodgy affair in the campaign, so you might need to do this one same as "Organ Grinder". Once you have the Boomshield equipped, down your enemy by either smacking them upside the head, or shooting them. Once they're downed, approach them and press the B button. You'll raise the shield and bring it down on their widdle noggin'. Do this ten times and you're done.
Smells Like Victory (10)
Torched 30 enemies
The Scorcher shows up a whole bunch in both the campaign and in Horde mode so pick one up and get to making a fresh batch of Crispy Critters, Sera style. If you still don't have it after making up a tasty dish of Wretch Flambe, take care of this one the same way you did "Organ Grinder". However you do it, make sure you keep laying the flame on once your enemy is downed as if they bleed out after bring burned, the kill doesn't count towards the achievement.
Variety is the Spice of Death (30)
Killed an enemy with every weapon
You can get this one naturally through the campaign, or through Horde mode, provided that you keep track of when you've killed enemies with the different weapons, but going the "Organ Grinder" route may be easier. For one, you don't have to worry about being saddled with a crappy weapon in the campaign. Second, if an enemy bleeds out, you don't get credit for the kill, and it's easy to shoot someone with, oh, I don't know, the Boltok pistol and you think that you got credit for the kill but instead they bled out and then you do the other weapons and the stupid achievement doesn't pop. Not that I have personal experience with that sort of thing.
Here's a list of all of the weapons in the game. If doing this one in multiplayer, weapon substitutions are your friends.
Snub Pistol (COG); Boltok Pistol (Locust); Gorgon Pistol (Locust); Lancer (COG); Hammerburst (Locust); Boomshot (Locust); Shotgun (COG); Hammer of Dawn (COG); Longshot Sniper Rifle (COG); Torque Bow (Locust); Heavy Mortar (Locust); Mulcher Chain-gun (Locust); Scorcher Flamethrower (Locust); Frag Grenade (COG); Ink Grenade (Locust)
For Ink Grenades, make sure that the poison ink kills the enemy and that they don't bleed out. Best thing for this is to either down an enemy and then toss an ink grenade at them, or toss one right after another.
Crossed Swords (10)
Won 10 chainsaw duels
Chainsaw duels are somewhat hard to come across in the campaign, as it's best not to let a Locust get close enough to where you have to duel them. Ditto for Horde mode. You can certainly use two controllers and the "Organ Grinder" method to get this one, but it's easier to enlist a friend. Start up a local multiplayer match, have each of you equip the Lancer, rev your chainsaws and then get up close and personal. You'll automatically go into the duel animation and once you mash on the B button enough, you'll make with the slice and dice and win the duel.