The Boar's Head Feast
Produce a Tavern Meal

Unlock the Tavern Kitchen. Build it and employ a cook. For the Tavern Meal you also need Cheese and Boars.



The Gabelle Gambit
Enact all seven Taxes

Residential Tax
Milling Tax, Mining Tax, Smithing Tax, Drinking Tax (Labour Progression Path)
Tithe Tax, Merchandise Tax (Privilege + quest for Merchandise)


The Exchequer's Haul
Collect 5,000 Tax Coins in a single month.

Get more villagers and the provide for them. When you have around 500 villagers, you've done it.



Teloneum
Enact the Merchandise Taxes Privilege (Challenge rules or harder)

You can unlock it by a random quest. You will get a visitor with some needs. He appears quite early.



The Big Cheese of Cheddar Palace
Build a Great Hall worth 100 Labour Splendour

Straight forwards, build a Great Hall and spam decorations.




One of the People
Complete the Aspiration: A Prestigious Burg

Complete all main quests. You will need to host the Seneschal and his entourage who have the additional needs of Refined Food x3, Goods x2 and Luxury x2 and Entertainment. You will get additional quests that increase his happiness. Prepare enough beds in the inn, 40 should suffice for all visitors.

The Burgomaster
Complete the Aspiration: A Prestigious Burg (Challenge rules or harder)

Complete all main quests with Challenge rules.

The Way of Saint James
Host a Pilgrim

Build the Hospitium and wait for visitors, some will be pilgrims.




Ora et Labora
Produce a Monastic Meal

Build the Monastic Kitchen. Any meal will suffice.




Via Commercii, Via Dei
Trade 500 resources with Kinstone Abbey over 30 days

Level up your trade route and produce many resources for trade.




First Growth
Enact the Seal of Prestige Privilege (Challenge rules or harder)

You can unlock it by a random quest. The steward of the bishop will visit alone and has the additional needs of Monastic Food x2, Luxury x2, Vineyard and Winery. The quest will appear after building the Abbitial Church or hosting the Abbess.

Divine Architecture
Build an Abbatial Church worth 100 Clergy Splendour

The Abbatial Church is the last building in the Clergy Progression Path. Build and decorate it.



Prior of Plenty
Complete the Aspiration: A Prosperous Priory

Complete all main quests. You will need to host the Abbess and her entourage who have the additional needs of Monastic Food x2, Abbatial Church, Apiary. You will get additional quests that increase her happiness. Prepare enough beds in the Hospitium, 40 should suffice for all visitors.

Abbey of Abundance
Complete the Aspiration: A Prosperous Priory (Challenge rules or harder)

Complete all main quests with Challenge rules.

Marching On!
Form a Company

The first you do after building a Castle.




Sound the trumpets!
Complete a military mission

Open the Mission List in the Army Menu. Send your company on a mission and wait.



A Legend is Born
Promote a Soldier to Knight

The tiers of a militiaman are 1. Militiaman, 2. Soldier, 3. Knight. Train your companies by sending them on mission to level them. You can promote on level 4 and 9.


Honi soit qui mal y pense!
Successfully host a Company of Royal Knights

They come visiting after some time. They have the needs of Refined Food x2, Quarters Housing, Luxury, Service, Goods and Entertainment.



Veni, Vidi, Vici
Complete a Military Mission of Grade 3

You can unlock higher graded missions by completing the previous grade. You will have company requirements for each mission, so create a company for every weapon type.


Salt Fish and Pottage
Stock 400 War Rations

Build some Provisioners to create War Rations. Stockpile and wait. Provision will be needed for higher tier missions.



Centuria
Have an army of 100 Combatants

Build Encampments for 100 combatans. Fire your people from their jobs to enlist them for military work. You can reload afterwards.



Montjoie! Saint Denis!
Deploy a Battalion to a Military Campaign

This is the last mission you unlock you will need tier 3 companies with at least one knight each. Four companies are required: sword, spear, bow and any.



Against All Odds
Complete a Military Campaign (Challenge rules or harder)

Wait for the Military Campaign to finish. It will take 2 month. Their will be some subquest you have to complete, stock your Kingdom Influence to be safe.



Fit for a King
Build Private Quarters worth 100 Kingdom Splendour

You can unlock the Private Quarters in the Kingdom Progression path. Build and decorate them.



Iron and Oak
Complete the Aspiration: A Mighty Stronghold

Complete all main quests and other requirements. You will need to host the king and his entourage who have the additional needs of Refined Food x2, Luxury x2, Goods x2, Service, Great Hall, Entertainment and Training. So this will be expansive. You will get additional quests that increase his happiness. Prepare enough beds in the Private Quarters, The king an his man will need 13.

Steel and Stone
Complete the Aspiration: A Mighty Stronghold (Challenge rules or harder)

Complete all main quests and other requirements with Challenge rules.

Expansion
Purchase your first Territory

Probably the first achievement you get. Just play the game a bit.




Your First Church
Build a Rustic Church

You can unlock the Rustic Church in the Clergy Progression Path.




Spices of Life
Start a Trade Route

Build a Warehouse or a Granary and store a resource. Open the resource tab in the Royal Book. Click the arrow up or down right of your stored resource. The number shows you the target amount in your village inventory.


Merchant Prince
Upgrade a Trade Route to Level 5

Trade a lot of resources and unlock the next levels with mandates.




Room for One More?
Build a Workplace Extension

The first one will probatly be the Lumber Camp or Storage Improvement in the Common Progression Path. After unlocking it edit your Lumber Camp or Warehouse and add an extension.


Hammer Time!
Produce Tools

Unlock the production line in the Common Progression Path and build the required buildings.



Serf and Turf
Tax a Villager

Build a tax office, you can unlock it with the Manour House. You can get the tax tab in your Royal Book by clicking on your coins.



Just this once... I promise!
Earn Coins from the Levy Mandate

Open your mandates by selecting it in your Great Hall, the levy is the second option.



Happy as a Lark
Have a Villager with 150% Happiness

Decrease taxes, fulfill all needs and do some quests and you will properly get this one while completing the other achievements.
Otherwise build a City Belfry.


Bathing in Gold
Amass 10,000 Coins

Build a big Treasury and play the waiting game.




Calling a Ban
Enact an Edict

Open your Royal book and enact any Edict in the Edicts and Privileges tab.




Ye Noble Black Prince!
Obtain the Black Sheep

Build a sheep farm and create a big pasture to spawn as many sheep as possible.



A Fief Too Far
Own 30 Territories

Buy many territories. You can safe up, buy them and reload.




Feudal Overlord
Own at least 30 Territories (Challenge rules or harder)

The same as the previous one, but upkeep is higher.




A Growing Village!
Have a population of 100 or more

Accept villagers and give them jobs.




A Thriving City!
Have a population of 500 or more

Accept more villagers.




Civitas Maxima
Have a population of 500 or more (Challenge rules or harder)

Accept more villagers, but harder.




Hanseatic Spirit
Upgrade all Trade Routes at least once

After completing your chosen Aspiration unlock the Administration buildings of each path, build and decorate them for more Splendour to increase your influence cap. Unlock the remaining trade routes when you have the influence.

King Midas
Earn 15,000 Coins within 30 days

Probably the hardest achievement. Prepare to play the game for a long time.
You can earn the most gold by taxes (Labour), trading (Clergy) and missions (Kingdom). The levy mandate can give you the last push.

Thanks to @Rhym for his comment regarding this.

We did it with a 500+ pop Prestigious Burg aspiration game, on challenging with no bailiff splendor for higher trade prices, with these tactics:

Prep:
- Save your game, you're probably gonna be loading afterwards.
- Have a large and full treasury so you survive the prep phase without going bankrupt. I had 10k, but depends on your economy.
- Enable all taxes
- Stop collecting taxes until they are capped out for each location. Have enough tax collectors to collect quickly afterwards. I notice they still collect enough if they're paused. You need to fire them.
- Save up resources for sale. It doesn't have to be insane amounts since the traders can only buy so much. It doesn't have to be all of them either. Include some trade goods from the clergy.
- Missions are not much worth it, so ignore them.

On day 1 of the month you're trying for the achievement:
- Start all tax collecting again.
- Stop all goods purchases.
- Set all resources to sell at above 0.
- Enforce a levy every time you can. Since it's a 7 day cooldown you might want to wait for near the end of each week.