Here is a brief rundown on the level ratings you can get.
The highest rating you can get is a Special Perfect Double+

Perfect is when all 100 gold bars AND all boomboxes have been collected in a level.
Special is when all 25 gems AND 100 gold bars AND all boomboxes have been collected in a level.

A + (plus) is added to either of these scores if a bullseye is scored in the extra micro-game at the end. You only get to the micro-game if all 100 gold bars are collected, in either Run mode.

Keeping that in mind:
Perfect+ is when you collect all 100 gold bars and then get a bullseye on a Gold Run.
Special+ is when you collect all 25 gems and 100 gold bars and then get a bullseye on a Gem Run.

Therefore, the highest rating is achieved by completing a level twice: once with a Perfect+ and once with a Special+.

As mentioned, you need to score a bullseye to get the maximum rating. You only get one chance at this, so it may be worth practicing this on the early, easier levels.
Personally, I fire the cannon on its very first climb up and I seem to get it pretty consistently. It's difficult to explain, but give it a few tries and hopefully you'll get used to how it works.

World 1
This one is in 1-3: The Meaty Treaty. Right near the start, take the alternate path to the right and you'll find the VHS.

World 2
This one is in 2-8: Parallel Peril, just past the checkpoint. You'll need to jump down a cliff immediately and you'll find the tape if you jump down earlier than the gold suggests.

World 3
This one is in 3-4: Cart Issues. Once you've unlocked the gem path, run it and you should easily be able to see it. Thanks Psychopath for the exact location of this one.

You're Perfect!
Get a Perfect!


Better Than Perfect!
Get a Perfect+!


How Special!
Get a Special!


More Special!
Get a Special+!


Especially Perfect
Get a Special AND a Perfect in any one level!


You're Incredible!
Get a Special, a Perfect, and one Bullseye in any Level!


You're Unstoppable!
Get a Special+ and a Perfect+ in any Level!


If you don't understand any of the requirements for these achievements, the previous section Introduction should cover them.

A New Challenger
Checkpoints? Who needs 'em!


Complete a level after skipping a checkpoint by jumping over it. I would recommend doing this on one of the first levels to make it easier on you.

A Perfect World...
Get a Perfect in each Standard Level in any World!


Just like it says. Do the same as You're Perfect, but for every level in a single world.

A Special World...
Get a Special in each Standard Level in any World!


Just like it says. Do the same as How Special, but for every level in a single world.

Unbelievable!
Make it to the end of an impossible Level!


Unbelossible!
Finish all 9 Impossible Levels!


Impossible Levels are unlocked when a certain total gold amount is reached in each world. These can be difficult. It's in the name.
This is actually a little easier since the difficulty update. You can change any difficulty settings as long as the score multiplier does not fall below 100%. I recommend extra checkpoints, also removing gold and gems (if you don't need them) can be less distracting and doesn't change the multiplier.

Unpossible!
Collect all Golds and Gems and finish an Impossible Level!


...good luck!
The difficulty update made this a fair bit easier. The score multiplier doesn't matter at all, so you can turn off absolutely everything (except gold and gems) if you're having trouble with this.

Bah Hamburger!
Defeat Foodland's Boss!


This is the boss at the end of the first world. Just dodge the rockets, pick up the sausages and send the meat balls back at him.

Parliament is in Recess...
Defeat Spookyland's Bosses!


These owls are at the end of the second world. Just dodge the various obstacles then perform the required move on the gavel. This will be a trampoline, then a kick move, then finally a ground slam.

A Fish out of Water
Defeat Machineland's Boss!


This little fishio is at the end of the third world. Dodge his spinning attacks and jump on top of him using the jump pads.

John Jacob Bonkenheimer Schmidt
Bonk the Timbletot!


After beating all other bosses, the Final Battle will be available from the World Map. Beat this guy to end it all.

Fall came Early this Year
Defeat The Great Outdoors' Boss!


Spider? I Hardly Know Her!
Defeat The Caves' Bosses!


Who's your Bone Daddy!
Defeat the Jungle Ruins' Boss!


These bosses are all found in the last level of their respective Retro worlds.

Puppet Master
Collect all the Narrator's Puppetshow Puppets!


There is generally at least one of these per level. You will unlock this the second you collect your last puppet. You don't even need to watch the puppet shows, but you should because they're great.
Any puppets you pick up on a run will be immediately saved, so you never need to recollect it, even if you get bonked or exit the level right after.

Setting the Standard
Get all the Gildans!


Gildan is a fancy word for coin. Coins are found in the retro levels, five in each one to be exact.

Who Dat!?
Invite a friend to the BBQ!


Unlock your first character by completing a Hero Quest. Hero Quests are completed by collecting items for a certain NPC in various levels.

A Full Roster
Get all your friends over for some GRILLINS!


Unlock all characters in the game. You unlock a character by completing a Hero Quest. There are three quests per world. To find NPCs, you can see them on the world map right next to the level you can find them in.
There are 11 in total, including the starting two. Here's a list:
CommanderVideo
CommandgirlVideo
Unkle Dill
Awnty Rewty
Dave
Frank 'N Stein
The Narrator
Eddie Riggs
Sadbot
The Defector
Shovel Knight

I Feel Pretty
Buy a bunch of Accessories!


You need at least 5 out of a possible 16.

Oh So Pretty
Buy a bunch of Capes!


You need at least 10 out of a possible 24.

A Diverse Wardrobe
Buy a bunch of new digs!


You need at least 15 out of a possible 64.

As these three achievements suggest, you purchase the items for these at a shop, which you can find one of in each world. These items cost you a combination of coins from retro levels and gems from gem runs of standard levels.

Destroyed
Unlock EVERYTHING!


I'm not totally sure what this requires yet, but I would guess all Characters, Levels (including Impossible and Retro Levels) and Items from the shop.
It was confirmed that you will need to get a Special Perfect Double+ on every possible level for this achievement. For clarification of what that means, check the Introduction section at the top of the page.

Is there a Duck in Here?
Better out than in...


When you get a perfect and shoot out of the cannon at the end of the level, hold the button down and you'll get this one. This can be done with any character. Thanks to Culex64 for finding this.

A Pecret Sassage
Go exploring in the Retro World. Maybe you'll find a cute secret?


This one is in the third Retro world, Jungle Ruins. Start level 6-2: Legend of Tom Braider and drop down below the starting area as soon as you can. There are some hidden panels that you can enter that are slightly darker than the rest of the ground.
Just run into this area to get this achievement.
There's a similar area in 6-4: Ruined Ruins. There may be some other areas that will count for this achievement, but you only need to find one.

Did You See That!?
Witness something creepy.


This a random event that occurs while using Charles Martinet as a playable character. It can happen anywhere that Charles Martinet is visible on the screen including the shop, level select, score screen as well as during levels.
You need this event to occur 13 times. You can check your progress at the very bottom of the stats page, but if you play as Charles since you unlock him you won't need to focus on going for this achievement or you can idle with him if you're done with everything else.

What's This?
Head off the Beaten Path...


For this achievement, you need to collect at least 1 character medallion. Read below for locations.

Numismatist
Got 'em All!


Get all aforementioned secret character medallions. These are found in secret areas throughout the standard levels. There are 11 in total (one for each character) and I will outline locations below as I (or Culex64) find them.

Here is a brief outline, with in-depth locations further below:
- 1-2: A Briny Solicitation
- 1-5: For Cheddar or Wurst
- 1-6: Run A Fowl
- 1-8: Fishing For Condiments
- 2-4: Doll Hollow
- 2-6: The Wording Way
- 2-7: Bound By Law
- 2-8: Parallel Peril
- 3-2: Subpar Contraption
- 3-6: Maniacal Mechanical
- 3-9: Heavy Metals

In the level 1-2: A Briny Solicitation you will come across a pit shortly after finishing the plane segment.
Use a ground slam to make it down the pit, then dodge all the obstacles.

In the level 1-5: For Cheddar or Wurst complete the first minecart phase. Shortly after there will be a few enemies and a boom box. Switch lanes to the right just past the boom box to access the path to the medallion.
A short platforming section later, you will find it.

In the level 1-6: Run A Fowl, you will come across this trampoline on the Gold Run past the checkpoint.
Jump over this one and clouds will appear under your feet to help you reach the hidden one just past it.

In the level 1-8: Fishing For Condiments, play through to the checkpoint. There will be a car section shortly after. After the first spiky totem, stick to the right and drive right towards this boarded up wall:
This wall will open up and you'll find the medallion at the end of a very narrow snaking corridor. It's in the air at the end, so be ready to jump up for it.

In the level 2-4: Doll Hollow, play through to the checkpoint. Immediately after, follow the path on the right and then switch to the left to a path on the corner shortly after. On this path you'll enter a portal inside a doll's head.
On the other side, the medallion is after a short platforming section.

In the level 2-6: The Wording Way, play through to the checkpoint. Keeping on the normal path you'll take a jump pad. When you land there will be two kickboards. Around the second one you will be able to switch lanes (to the right) to access the secret area.
For reference there is a large question mark in the background, but you have to switch lanes before you get up to where you can really see it properly. This one's rough because it's not near the start or a checkpoint and has a pretty tricky platforming section.

In the level 2-7: Bound By Law, play through till you get to the checkpoint. Immediately after the checkpoint there will be a split path.
There's actually a secret third path above the other two which you access by moving to the left. You'll have to jump over two small wooden platforms to get to the medallion.

In the level 2-8: Parallel Peril, there's a car section at the start of the level. During it you'll want to drop down the very first gap that opens up.
You'll just have to move left and right to dodge the obstacles down there.

In the level 3-2: Subpar Contraption, follow the Gold Run path till you get to the first split in the path. After going towards the gold path, switch over to an extra path in the middle.
At the end of here, you'll want to do a massive double jump while trying to get the maximum distance. Get ready when you land because you won't have much time to react to a quick jump then slide. Luckily it isn't too far from the start of the level.

In the level 3-6: Maniacal Mechanical, after the checkpoint, you will enter a plane segment. After passing through two circular doors, you'll enter this room with black and yellow platforms either side of you.
Immediately after these walls end, dive straight down to find this medallion.

In the level 3-9: Heavy Metals, play through till the last boom box (EXTRA). After taking one jump pad, skip the next one and instead do a big double jump toward the white wall there.
It will collapse and you'll play through a pretty small section. You'll have to take the jump pad in this area to get the token.