Vex Clock has many things to discover and requires multiple playthroughs to unlock everything, so this guide aims to list the achievements, collectibles, and cosmetics that you can aim for. BEWARE OF POTENTIAL SPOILERS.
NOTICE: ‘Endings’ section is incomplete until I figure out how to get them all
Base Unlocks
Each act (and the tutorial) has an unmissable achievement for completing them. Each area also contains Computer Chips, a collectible item hidden throughout used for unlocking costumes.

Complete the tutorial

Complete Act 1

Complete Act 2

Complete Act 3

Complete the first half of Act 4

Complete the second half of Act 4

Complete Act 5
When you first play the game, one of the settings available is difficulty- which each have their own respective achievement:

Beat the game on Easy difficulty

Beat the game on Normal difficulty

Beat the game on Hard difficulty
These achievements unlock based off the highest difficulty chosen, meaning that beating the game on Hard mode will unlock the Easy and Normal achievements as well. This can be useful if you want to reduce the amount of time achievement hunting.
Important: While you can change difficulties mid-run via the game options, you cannot get the Hard achievement if you ever have the game set to anything except hard difficulty for that run.
Also in the game settings is a toggle to show the title and artist for each music track that plays in the game, giving a nice insight into the soundtrack as well as this achievement:

Toggle the music credits feature. I hope you enjoy our music!
Along with other applicable achievements, there’s things you can unlock for simply beating the game. The two gamemodes will unlock a new option and 5 costumes for the characters, whereas the modifier only grants an achievement.

Beat the game in New Game Plus

Beat the game with the randomizer modifier enabled
There’s also an achievement tied to beating the game but not tied to any particular setting or mode, being:

Beat the game without dying once on any difficulty
For this achievement, I recommend playing NG+ on Easy mode; NG+ lets you start with all abilities unlocked, and Easy mode both makes it easier to heal and to take down enemies.
Remember: Dying once means you have to start a new run entirely!
Objective/Collector Achievements
Computer Chips are the game’s main collectible, existing in every act and zone. The first one you’ll be prompted to collect in the Tutorial, which also grants:

Collect your first Computer Chip
Afterwards, when meeting a new milestone you’ll be rewarded with costumes and an achievement. The milestones, starting at 10 chips, are initially 5 chips apart- but become 10 chips apart once 30 chips are collected.

Collect 10 Computer Chips

Collect 15 Computer Chips

Collect 20 Computer Chips

Collect 25 Computer Chips

Collect 30 Computer Chips

Collect 40 Computer Chips

Collect 50 Computer Chips
If you’ve gotten all the Computer Chips and have gotten the costumes from beating the normal game and NG+, then you earn this achievement:

Collect all 25 Costumes
Two achievements can be gotten with the same mindset of: if it lives, kill it dead.

Kill 439 enemies

Kill every crow you see, leave no survivors
While John Vex Clock will happen naturally as the game tallies kills over runs, ‘You Monster…’ is a little trickier; The crows are much less plentiful, and this must be done in one run (unlike other achievements in this category). If a level has crows, you’ll hear ambient cawing- keep an eye out! There’s 32 total crows to kill, often perched in innocuous areas, and a level with crows in it will have 4 of them. There’s 4 in the tutorial, 4 in the Southern Slums, 4 in Fishly Land, 12 in Siren City, and 8 in the castle gardens. After you’ve killed all crows in the area, you’ll hear a deep, echoing caw.

Help Berry with his vending machine problem
There are 5 vending machines tucked away somewhere in the first zone of Acts 1-4. Attack them a bunch until the soda pops out; After all 5 machines have been destroyed, continue playing normally. You’ll meet Berry the Bear on the bridge in Act 5, zone 1, granting the achievement.
Endings (WIP)
Throughout the game, you’ll come across personal missions that change who in your party you can use for a particular zone- these missions are part of the bonding system. Complete enough personal missions with party members to build a strong bond, and the ending will vary depending on who’s closest to you. Each ending has its own achievement.

You Resolved Everyone’s Problems
Earned by bonding with Lilian and Rosa

Without luck, everything falls
Earned by ???

Underestimated the secondaries
Earned by bonding with Lilian and Katrina

Love conquers all
Earned by bonding with Chloe and Rosa

A new queen takes the stage
Earned by ???

You violated the law
Earned by bond with Chloe and Katrina

A rose stained by darkness
Earned by ???

The cunning cat reigns victorious
Earned by ???

Regardless of bloodlines, the cycle continues
Earned by not getting close with anyone