This Guide May Be Slightly Out-Of-Date! Any And All Feedback Is Welcomed <3
I Will Also Probably Be Unable To Answer Any Questions You May Have As I Made This Guide Years Ago And Do Not Remember Every Detail About This Game.

Just To Clarify, There Is A Timed Achievement, But You Shouldn't Be Too Concerned.
This Method Worked For Me, If, For Instance, You Run Out Of Dynamite, You Will Have Plenty Of Time To Return To The Surface.

Getting all the achievements can get to be a bit time consuming, however, I am here to show you that it is possible to get all (but one) in a single run. It helps if you have played the game once through already so you know the basics and how the game is structured.
You also do not need to rush through the game to get the Master Prospector achievement either. Take your time, enjoy yourself.

The biggest time saves will come from skipping all the dialog and a dynamite jump (more on that later). Apparently You Can Skip Cut-Scene's Now... If So, Do That :)
If you happen to die at any point during the run and you are on the "You Are Dead" screen, alt + f4 out of the game. This will take you back to the last save and won't count the death.

First things first, start a new game. A small intro will play, but you can spam the enter key and spacebar to wizz through it.
Retrieve your pick and dig down to the lever.
Enter Archaea. As soon as you do, you will get the first achievement:

Scratching the Surface - You found Archaea!

Now, before you go to the surface, dig up 8 Trashium. Once done, reach the surface and sell. This will award you with your second achievement:

Trash Man - Find and sell 8 Trashium ores at the same time.

Talk to the Old Robot and buy the pick upgrade.
Go back into the mine and dig and explore ALL caves in Arachaea. Feel free to get any secrets but don't waste time searching for them [I'll go into detail at the end and where to find them all]. Only mine ore that is relatively close.
Have a full inventory? Sell your items.
The only upgrades you need to buy from shops are pick upgrades, pouch upgrades for now. Light upgrades are useless.
Always keep a teleporter on you and don't feel threatened to use them (I used at least 10 in my run!).
If you come across any blue orbs along the way, be sure to collect them.
Once you have the drill upgrade, the pick becomes useless. You should always have enough water to constantly use the drill. It is much faster, so prioritise on drill, pouch and water upgrades. Also buy armour upgrades to increase survivability.

Be on the lookout for any Trilobites that are still asleep. This will help to get:

The Good - Defeat 10 Trilobites before they wake up.

While we are in Archaea, in one of the unnamed caves is a few 2x2 Boulders and two mecha Turtles. Re-enter this cave and dig out the space directly underneath the rock so all the boulders fall. If you time this right and do it ten times then you will be one step closer to a perfect game:

Dig Dug Skills - Crush 10 enemies with falling blocks.

Once you have done everything in Archaea, enter the Old World!

As soon as you discover this dark and gloomy place, this should pop up:

Mythical Creatures - You found the Old World.

Go back to the surface if you have anything to sell and buy the upgrades you need.

You will be skipping all the unnamed caves for now.
Enter named cave one and receive the dynamite upgrade.

Once this has been achieved, return to the surface and enter Archaea again [Sell if you have to].
Enter named cave one. On the very top is another entrance, we want in.
Position yourself under the left most bit. We are going to do a dynamite jump. Make sure you have health or you will die.
Here's what you have to do:
Throw a stick of dynamite directly below you.
Prepare to do a steam jump.
As soon as the dynamite leaves your hand, count to three and JUMP.
This will send you flying to the roof of the cave, and you should be on the top most part with a door, enter it.

In the room in-front of you will be two mecha Turtles. Wait until one of the is in the water next to you and kill it:

Turtle Soup, Yum! - Defeat a turtle submerged in water.

Get the diamond, orbs, and secret with four dark matter to snag a bit of cash.
Make sure you have three uses of steam jump.
Use one to jump up through the gaping hole above the boxes. DO NOT jump too high or the lasers will activate.
Use steam jump again to launch yourself up and past the lasers.
On your left should be a turret and two Trilobites, kill the bugs and get the secret orb on the far left.
Steam jump up to the weak block to force the "falling" block to go right.
Stand on it once it is still, mine the blocks and destroy the fan/generator.

Return to surface if you have to sell items AND buy dynamite as well as upgrades.
Keep mining as you would usually do, get blue orbs, etc.
Enter Named cave two and get the Steampunch upgrade.
Enter Named cave three and there is a secret directly to the left with a lever in it. Pull it and carry on with the cave as normal.
Once to the bouncy mushrooms, to the very top right most one is a secret, stand on solid ground and throw dynamite at it. This will cause it to bounce and blow up the mine-able block. This will reward you with a few more orbs.
Get the Fall Dampeners.

In the same cave, there is another huge, long hole above you.
Dynamite jump and reach the top.
Dig out all the precious ore and head left.
Try not to take damage, get the orbs and head towards this new entrance.
As soon as you enter, there will be a Shiner sleeping. Kill him while he is sleeping. Go back through the door and kill him again until you receive:

The Bad - Defeat 10 Shiners before they wake up.

There should also be orbs above him.
Head all the way to the left and there are a few boxes, destroy the bottom one, jump on the other two and mine the one you are standing one. Steam punch into the wall and there should be another secret.
Carry on the cave as normal.
When you get to two sleeping Shiners, kill 'em and dynamite jump for the last time. Complete the rest of the cave and destroy the fan/generator:

Sequence Breaker - Destroy two generators before entering Vectron.

Once done, head down to Vectron...

Just for entering this place:

It Goes Deeper!? - You Found Vectron!

Keep an eye out for sleeping Mecha Trilobites. Kill 10 of them:

The Ugly - Defeat 10 Mecha Trilobites before they wake up.

This is where it gets fun. Make your way down to the first Named cave of Vectron and get the metal detector upgrade.
This is where you mine, and explore EVERYTHING in Vectron.
Mine all the ores and orbs and explore all caves.
Once you make your way quite far down Vectron, you will start to encounter Uranium.
Try and get EIGHT in your inventory and go and sell it for this:

Half Life 8 Confirmed - Find and sell 8 Uranium ores at the same time.

Once you reach the Boss room, do not enter. Simply sell ores and keep doing so. By now, you should have bought at least ONE upgrade from every shop:

Support Your Community - Buy from all of Tumbleton's shops.

A small ways down from the Boss room should be the bottom of Vectron:

Rock Bottom - Reach the bottom of the world.

Around here should be A LOT of Dark Matter. Try and get four stacks (16) of this and sell it:

What's the Matter? - Find and sell 16 Dark Matter ores at the same time.

Alright, we are coming to an end. Not much to do now. If you are worried on time, go to the main menu and check. You should still have plenty of time (I finished this with more than a hour left).
This is where you go back to Archaea and get all the blue orbs you can find (just as a cushion).
Enter the Old World and enter the un-named caves and get the rewards from them. Mine out the rest of the Old World of blue orbs.
Finish off mining out Vectron and go back to the surface.
By now your total cash should be well above 20000 gold. This achievement will pop at the end of the game:

Two Kinds of People - Complete the game with more than 20000 gold collected in total.

You should also be able to buy the rest of the upgrades, and I mean buy EVERYTHING from every vendor. This should secure you with two achievements:

Fully loaded - Maximise All Upgrades.

You should also have spent a lot of blue orbs, hopefully totalling 150 (I finished with 200+) or more. This will also pop at the end of the game:

Shine on, You Crazy Round Thing - Complete the game with more than 150 orbs collected in total.

Now, all that is left is the secrets and the Boss!

Most secret areas aren't "really" hidden but they do reward you with a few goodies. Even in a casual play-through, you most likely got most of these:

The first secret is in the very cave. Enter the generator room and it will be next to the boxes. It contains 4 dark matter.

The second secret is in the same room. Jump up to the two Trilobites and laser. It will be on the far left.

The third is in the last cave of Archaea where the drill is located. When you enter, it is visible above you. Make your way to the left of the Cog and mine through.

The next secret area is in the an un-named cave with a lot of bouncy mushrooms. Make your way to the top right most part of the cave and mine out the block, revealing the secret.

Now make your way to the cave where you unlocked the steam punch (named cave two). To the right of the cave should be a shiner sleeping on three boxes, kill him, mine the boxes and get the next secret.

The following is a stupid easily secret. When you enter named cave three (where you unlocked fall dampeners), it will be directly to the left with a lever in it.

While you are in this cave, head towards the right until you reach the bouncy mushrooms. Head up. In the top right should be a few mine-able, but inaccessible tiles. Throw a stick of dynamite on the mushroom and it will bounce and destroy a single tile.

Carry on to the upgrade machine. When you reach it, on the left wall should be a mine-able tile. Use the steam punch to get to it.

The last secret area of The Old World is in the generator room. Head up and go left towards the boxes. Destroy the bottom box. Jump on the other two and mine one. Use steam punch to get to this secret.

Vectrons secrets are a bit more hidden. The blocks don't look destructible. They are silver on the top and black on the bottom with two thick wires connecting them. Any time you see these in a cave, you can destroy it.

The first secret is in the room where you get the metal detector upgrade (named cave one). Just before you enter the "maze" part of the room, on the far right is a secret tile in front of a few fans.

Head towards the upgrade machine. Jump on the top right floating block and use steam punch against the right wall.

Keep going along the path. When you reach a hole, drop down. It will be a small jump to the left from where you landed. It will be on the roof next to some respawnable tiles.

The next few secrets are in the second named cave where you unlock static jump. Head to the top of the cave just before you would have to use a steam and static jump. Use steam punch against the wall on the left near respawnable tiles.

Head directly to the right of where you used the steam punch. You should notice a water source. Just mine the block left of the small blue light.

Once done, jump down the hole to the right and you will be able to see a Cog. Mine the block under it with steam punch.

Before you leave this cave, to the right of the entrance should be two laser. Destroy them. On the surface they were on, the left wall will contain another secret.

The next secret area is in a cave with two laser boxes next to the entrance with a Trilobite below you. Kill the Trilobite and under the laser box should be a secret on the right wall.

The penultimate secret is in the room with a lot of mecha Exploders. Make your way to the right of the cave and dig straight down.

The final secret area is in the boss room. Before you jump up between two lasers, there will be a visible Cog. Simply look for the specific block and the achievement is yours:

It's a Secret to Everybody - Find all secret areas in the caves.

Once you reach here, you will be in a room with three lasers. You should have max upgrades. Simply drill into one of the laser boxes until it breaks. If you run out of water, use your pick:

99 Problems and the Box Ain't One - Destroy a Laser Box.

Before you go up and towards the boss, the final secret will be visible. Destroy the one mine-able block and the achievement will be awarded as well as setting hp/water/light at full:

It's a Secret to Everybody - Find all secret areas in the caves.

Now the Boss. Skip the dialog and I recommend using the Steam punch to obliterate him. Two fully charged punches and a weak one will destroy a generator. Once done, jump on his head and pick him 12 times.
Do this another two times and congrats! You completed the game with all achievements.
After killing the boss:

Assimilated - Immortalization of a destructible steambot.

Once the final few bits of skippable dialog are over, the rest of the achievements should pop:

Two Kinds of People - Complete the game with more than 20000 gold collected in total.

Shine on, You Crazy Round Thing - Complete the game with more than 150 orbs collected in total.

Quicker than a Thumbleweed - Complete the game in less than 2 hours, 30 minutes.

Just Like Real Life - Complete the game without getting killed once.

Master Prospector - Get all four gold stars in the same play-through.

Okay, I lied. The final achievement requires you to dig up 15,000 tiles. Not difficult, but time consuming. I recommend playing through the game again. This should put you somewhere between 10,000-12,000 tiles mined. The way I did it was strip mine with the drill and steam punch. It makes quick work of every tile and with a little dedication, you will be awarded the final achievement for a perfect game!:

A Fistful of Dirt - Dig out 15,000 tiles.