I recommend finishing the game first, then coming back for any achievements you've missed. Finishing the game lets you replay all levels and visit sidequests that were once locked, while letting you keep all of your upgrades.

Ctrl+F to find a specific achievement.

Spoilers ahead!

Playing coop is a huge help on many Hell difficulty achievements, as well as mandatory for the coop ones, obviously.

However, if you don't have a buddy to play jolly coop with, a trick is to set Player 1 to use a gamepad and Player 2 a keyboard, then set up Xpadder (or similar software), binding the corresponding actions on the keyboard to the gamepad, so that when Thurstan jumps, so does Finn.

It's not perfect, as Finn has a different attack cooldown speed and doesn't stop if you press Grab without an enemy in front of you, but it still just works. The best way to play this way is to overlap the characters, so either make both of them hug a wall, or hold crouch / spam Grab and walk Finn towards Thurstan.

Here's the Xpadder scheme that uses the default keyboard setup for Player 2:

                                       

Can't be missed.

— Now you can throw daggers!
Achieved as soon as you level up once.

— Go corpse surfing
Achieved on an early part of Stage 1.

— Thurstan gets possessed
Achieved at the start of Stage 10.

— Collect all the questy things!
Achieved after obtaining the last quest item. Quest items are rewards to mandatory quests.

— You beat the game!
Achieved on story mode completion.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
Exclusive to arcade mode:
— Complete the Arcade adventure!
Achieved on arcade mode completion. Arcade mode features:

• No saves;
  
• A few sidequests skipped;
  
• Thurstan's weapons and Finn's equipment upgrades are dropped by bosses.

I recommend getting these at Stage 0, as the burned villager enemies are perfect for any kind of grab.

— Tear 'em in two!
Get near an enemy and press Grab, or Up+Grab.

— Throw 'em around!
Get near an enemy and press Forward+Grab.

— Deliver a punishing ball kick!
Get near an enemy and press Down+Grab. You may need to damage him a little bit before trying to execute this.

— Rip an arm off 'em!
Get near a medium sized enemy (like the one pictured below) and press Down+Grab.

                                       

— Set a hostage free
Two pairs of hostages can be found on Stage 0, one of them on the end of the forked path, near a checkpoint. Duck in front of one to free them.

                                       

— Use a continue on Stage 0
Start up a new game on Casual or Normal difficulty, die enough times and make sure to use a continue on Stage 0.

— Die in the water
Simply jump into a water pit. The earliest level for this is Stage 1.

— Beat a walking level without crouching.
Beat any non-autoscroll non-shoot'em up level without ducking.

— Pay your dues to the zombie priests
On either Stage 6 or during Chapter IV's sidequest, get to a zombie priest and perform a ball kick (Grab+Down) on him. Doesn't work on nuns, hmmm...

— Get hugged by a crawler
On Stage 14, let the bubble-egg pop to release a crawler, then let it grab your face fondly.

                                       

— Reach the max level!
Max level is 13. I got this as soon as I got to the main menu after finishing the game.

— Get the Cock Puncher
Can be found behind a destructible wall on Stage 2. Check out All of the Stuff!!! below for more info on this.

— Bought the 6-way magic daggers!
Confusing one... despite what the description says, you gotta buy the 5-way magic daggers, the last upgrade for Thurstan. You'll need 40,000 gold for this. After buying it, the achievement will pop up as soon as you start another level. I recommend getting only the first upgrade (3-way normal daggers) then saving up your money to buy this upgrade later on.

— Get all the pointy things!
Obtain all 6 weapons in the game, being them located behind some destructible wall on certain levels, or at the end of a sidequest. All weapons, except for the sword, are only obtainable via optional sidequests, so make sure to visit any sidequests that you haven't yet before proceeding to a new level, as sidequests are set as inaccessible if you beat the stage it's related to, until after you finish the game.

• Cock Puncher: To the left of the starting point of Stage 2, behind a wall.


• Sword: Dropped from the first mandatory sidequest along with the Golden Egg quest item. If you somehow miss picking up the sword, through the whole Stage 2 you'll find sheathed (or bronze?) swords resting on the ground -- simply pick them up and a common sword will be added to your inventory. It can also be found on the ground of Stage 10.


• Mace: Can be found inside a crate in a stairs section of Stage 4.


• Chain Mace: As soon as you pick up the rosary, thus unlocking the Stage 6, check out Chapter III (the island to the right). There should be an optional sidequest that unlocks the destructible wall leading to the Chain Mace inside the Stage 6. This weapon is guarded by quite a few enemies, and 2x minibosses at the end.


• Blue Mace: Behind a wall on Stage 7, at the bottom of the level's first drop. It's guarded by a giant blue eye blob.


• Golden Sword: This one is also located behind a wall on Stage 7, at the bottom of the level's second drop, quite close to the first one. It's guarded by a few Ents.


— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
— Upgrade Finn's axes to max level!
Join in as player 2, rack up some money and buy the Axe upgrade twice; the achievement will pop up as soon as you start another level after getting the upgrades. The earliest shop is located right at the start of Stage 1. Only Finn can buy this upgrade, meaning you'll need at least 8000 (2k+6k) gold on him, not Thurstan.

— Skip the How To Play Video
After starting up a new game, skip the tutorial video.

— Revisit the How To Play video
The tutorial video can be accessed on the main menu (the one with 1P Play, 2P Play, Options, Credits) by selecting "How to play".

— The taste of true defeat
Die enough times to get a game over message and this achievement. When playing on casual or normal, let the Continue timer run out.

— Wow! 100% on all your saves!
Getting 100% on a save requires obtaining all weapons (check All of the Stuff!!! above for more info) and finishing the game. After saving it at 100%, save again on the other two slots, then quit to main menu to trigger this achievement.

— You like to live dangerously, no saves!
Finish the game in one sitting without saving. It's recommended doing this on either Casual or Normal difficulties, so that you may use as many continues as you wish. Note that this achievement cannot be obtained in conjunction with Really - I didn't cheat!!! (finish hell mode without saving), which means you'll need to finish at least two full playthroughs with no saves

— Play singleplayer on the day of love
Make sure your computer's date is set to February 14th (either set it manually or wait for the date to come up), start up the game and you'll obtain this one as soon as you kill an enemy.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
— Play co-op on the day of love
Same as the one above, but join in as player 2 before killing an enemy.

— Play within punching distance!
Play coop and simply overlap the characters to get this one. Playing alone? Check out the Playing coop mode, solo section on the top.

Having trouble with achievements from this section? Play coop (check out Playing coop mode, solo on the top).

— Beat the game on Casual difficulty
This can be done on a completed save: from the main menu, after choosing a savefile, go to the Settings and switch to Casual difficulty before starting, then kill the final boss on Stage 15 again.

— Finish Stage 0 in Hell Mode
Start a new game on Hell difficulty and finish the first level. In the options menu, Hell difficulty is only accessible if you start on it right from the start, so the trick above doesn't work.

— Hey! We thought you liked it rough!
Start up a new game on Hell difficulty, then try to buy an 1UP from a shop (you don't need to actually buy it -- thanks Noxis for the info!).

— You are an ARCADE MASTER!
Despite what its description says, you'll need to finish the story mode on Hell difficulty. This means no continues, but you can save as much as you want. Not as bad as the one below...

— Go through HELL with no saves!
Start and finish a new game on Hell difficulty without saving. Hell difficulty features:

• Higher bosses' health;
  
• Higher damage output from enemies;
  
• One-hit-kill hazards (e.g. falling fireball on Stage 0);
  
• Fewer starting lives;
  
• 1UP from shops are more expensive (15k);
  
• No continues.

This achievement must be done on story mode. Learn the enemies' behaviour and their spawning points, and memorize one-hit-kill spots as well. Patience is the key word here, good luck!

There we go. If this guide has helped you out, don't forget to rate it. And if there is any way that I can make it clearer, do comment.