This first section is just for the basic puzzle completion. To get these achievements, all you have to do is complete the puzzle that is associated with the achievement.

Puzzle 1 Complete
Hard A** Hooker


Puzzle 2 Complete
Some Tongue?


Puzzle 3 Complete
Chewy Dogz


Puzzle 4 Complete
Don't Smoke


Puzzle 5 Complete
Hell's For The Homeless


Puzzle 6 Complete
The Beauty Of Birth


Puzzle 7 Complete
Underground


Puzzle 8 Complete
Tea Party


Puzzle 9 Complete
Hawk Eye


Puzzle 10 Complete
Prep For Bravery


Puzzle 11 Complete
Dead Monk


Puzzle 12 Complete
Shekly Shot Eye


Puzzle 13 Complete
What's On Tv?


Puzzle 14 Complete
The Bride


Puzzle 15 Complete
Undeadz 4 Lyfe Yo


Puzzle 16 Complete
Killer Clown


Puzzle 17 Complete
Back To Back


Puzzle 18 Complete
Rasta Gonna Get Ya!


Puzzle 19 Complete
No Where To Run

As you progress through the game, your character levels up and you unlock new puzzles.

Level 2
Zombie Hunter


Level 3
Zombie Slayer


Level 4
Zombinator


Level 5
Zombliterator


As far as i am aware, these achievements are pretty standard and are unavoidable.
The only part I doubted is if you complete a level having pressed the "without zombies" button, if you will still level up the same. As the achievement titles almost suggest that your character is only leveling up from slaying zombies. Therfore, if you avoid the zombies do you also not level up?
Anyway... If you have any insight on this, feel free to comment and i'll add your findings in here.

UPDATE: A commentor has confirmed that they played the first puzzle with zombies and the rest without given how tedius they are. They still progressed the same, got the level achievements and unlocked the later puzzles.

This section includes one of the two tedious achievements throughout this guide.

These achievements are earned by killing a certain amount of zombies in the menu mini-game.

Zombie Hunter
Kill 50 Menu Zoms


Zombie Slayer
Kill 100 Menu Zoms


Zombinator
Kill 500 Menu Zoms


Zonihilator
Kill 1000 Menu Zoms


Zombliterator
Kill 5000 Menu Zoms


I read in a different guide whilst I was playing that the 5,000 achievement had to be done all in one go. From my experience this was false, as I killed 200 or so zombies in one game, then killed the rest in another game. The achievement unlocked once i got to around the 4,800 mark.

UPDATE:

To be safe and not waste any of your time, try to get 5,000 kills in one sitting without closing the game.

From comments I have received on this guide, it seems that there is more people successfully getting the achievement killing 5,000 in one go than people getting it by doing it in different games within the same play session.

Due to these people having experienced different from me, It's assumed that the counter in the game isn't all that great and to be safe, you should attempt the 5,000 in one sitting.
If you get so far and just really can't go on any longer, then I would advise you to not close the game, have a break and at least try for the remaining amount that you need towards your last game just to see if it unlocks (As I would think it's better to have a chance at it unlocking then having to retry the full 5,000 anyway).

This section includes one of two of the most tedious achievements in the game.

These achievements are earned by surviving a certain amount of real time in the menu mini-game.

15 Min Madness
Survive 15 Minutes In Menu


30 Min Madness
Survive 30 Minutes In Menu


60 Min Madness
Survive 60 Minutes In Menu


The 60 minute achievement, I won't lie to you, is mind numbing.

But if you are having trouble finding a stategy that is working for you, i'll explain to you what my stategy was.

First of all, position your character in the center of the map.
I actually had my character slightly off center, as i noticed that one side had brighter lighting then the other. The extra light can be useful when you are trying to keep an eye on where the zombies are coming in from.

Secondly, weapon. Once positioned near the centre don't bother with your weapon. Only use the grenades. Even then, i use the word "use" lightly. What i really mean is "spam". Spam the hell out of the grenades.

Next, perimeter. Whilst throwing grenades at zombies, try and keep a mental perimeter pictured around your character, whenever a zombie is coming too close pick them off next.
Don't be afraid to spam grenades even if they are too far for the grenade to reach. The grenades stay there for a fair amount of time before blowing up and zombies actually walking into your grenades picks off a lot more then you might imagine.

Once you have killed a fair few zombies you will start noticing that them blowing up kind of starts to form a circle of blood around your character. Start to use that as a perimeter, it helps a lot being able to see clearly when zombies are crossing the line. Especially when you are trying to keep an eye on a lot of them at once, you're peripherals will see the red better.

Lastly, don't underestimate or let your guard down. They may seem steady and manageable but they do slowly start to pick up in numbers. Granted, i ended up there almost twice as long testing out another achievement, but towards the end of my play through there was quite a few to keep an eye on.

These achievements are the few odd miscellaneous ones.

Nade Thrower
Blow 'Em Up


Use grenades within one of the puzzles to kill zombies. Given the 3 i'm assuming use 3.

Claymore
Trip 'Em


Use the trip mines within one of the puzzles to kill zombies. Given the 5 i'm assuming use 5 or kill 5 at once.

Chomp Chomp
You'z Got Bit


Get bitten within one of the puzzles by letting the zombies destroy your defenses.