First off, it is highly recommended to set the language of your game client to English. It has been confirmed from the developers that there are problems with some achievements not unlocking in clients using other languages.
Niche features an in game developer console which can be used during story mode without invalidating achievements. This makes all achievements a lot easier, however whether you use it for yourself or not is completely your decision. Regarding achievement hunting, I know that everybody has their own set of morales, however at least according to Astats guidelines, making usage of in game consoles is allowed, as long as no achievements are unlocked directly via command.
I will include some commands in my guide, which you can use in the console. The console can be opened in game through the hotkey F1.

Generally, there are a few commands which will make the very challenging story mode a lot easier and which I would highly recommend using if you are not up for spending many hours in the story. To use a console command, click on a creature in your pack, open the console with F1, and enter the command.

• increaseage -1000 (Creature will die after 1000 days, making it practically immune to old age.)
  
• changeberrycount 10000 (Increases food count by 10.000, so you don't have to worry about food collection)
  
• exploreallfields (Removes all high grass from the map)
  
• switchgender (Self-explanatory, especially useful if your pack is getting old and missing creatures of one gender)
  
• setgene value1 value2 (Sets a gene with value1/value2. If you want a homozygous gene, you can leave value2 empty. I will cover some gene types in this guide, for others, you will have to make intelligent use of the consoles suggestions when typing.)

Homecoming
This achievements requires you to complete the story mode by reaching Adam's Home Island. There already is an excellent guide covering this written by Baarko. Combining that with the above-mentioned console commands, it will take you a while but should not be too hard.

Family Reunion
Additionally to reaching the home island, you will need to have at least one creature with the Home Island Immunity Gene (which is the star instead of the letters A - K). You can achieve this through careful and selective breeding. Sadly, I have not yet found the value for the home island immunity gene, meaning I have found no means of setting it via console just before reaching the home island.

Archipelago
Rewarded for traversing the archipelago map, which is in total the fifth map you will play in story mode.


Globetrotter
Rewarded for discovering all island types, which are: Tiny Green, Peaceful Meadow, Grass Adventure, Archipelago, Burning Savanna, Rain Forest, First Snow, Overgrown Jungle, Swampy Hill, Grass Mingle, Whale Island, Oasis, Summer Mountains, Long Winter, Crossing, Sleepy Reeds, Jungle Gate, Deep Jungle, Frost Lands, Deadly Hills, Home Island. Yes, this is one of the harder achievements. The developers have confirmed that this is only unlockable in story mode, e.g. you cannot just open all maps in sandbox mode and net the achievement.
Important Note: It seems that some version started bugging the achievement and it will currently not unlock. Rei Onryou kindly found a fix which can be accessed here.[steamhunters.com]
Update from January 2023: According to a user comment by Slick, this has been fixed and will work in the game's current version. Not verified by me.

Root of the Problem
Destroy five roots of the vines growing across the Overgrown Jungle island (refer to the abovementioned guide for more information) and travel northwards.


Adam's Curse
During story mode, you will regularly encouter a bird of prey flying above you. If you have the flying ability, you can attack it and even kill it. Otherwise, you can ignore it, as long as you don't have a baby creature close to it, since it will kill these if it reaches the same tile, which is also the condition for this achievement to trigger. You should be able to get it through abandoning a baby creature when you see the bird of prey very close to your pack.

These achievement require you to breed a creature with specific genes. This can be achieved easily by setting the genes of both parents correctly, guaranteeing the desired outcome. Some achievements require you to rename the child after it is born.
I would highly recommend going for all these achievements in story mode. I tried in sandbox mode and could not get some of these to unlock there.

These achievements require you to survive for a certain amount of days in story mode. These can be picked up on the way whilst completing story mode and the gene collection achievements (next section).

Survivor
Survive for 50 days.


Still Alive!
Survive for 250 days.


Never gonna give you up
Survive for 1000 days.

These achievements didn't quite fit in into the other sections, so I'll provide detailed descriptions on each.

Eating Nemo
This achievement will require you to eat a clown koi. I found loads of these in sweet water from the fourth story island onwards.
Important Notice: The achievement did not unlock for me in the game's current version, despite many tries. It seems to be bugged, however the developers have stated on the forum that they don't think the bug is on their side. However, when I opted into a previous release (1.1.4), it worked for me. Try that in case you are encountering the same problem.

Harem
Have one creature of one gender and at least five of the other. Can easily be achieved using switchgender.


Two for the Price of One
To be rewarded this achievement, you will have to give birth to twins. Having the high fertility gene on both parents (setgene fertilityhigh) increases your chances. Other than that, you'll just have to be patient.

Copy / Paste
Same procedure as the achievement above. You'll just have to be patient.


Sweet Sweet Love
Literally unmissable.


Spring Fever
You'll need a large pack for this. Gather them all and mate away! Note that a failed mating attempt counts, so having low fertility is helpful here.


Ice Age
I unlocked this achievement by discovering not five, but six prehistoric genes in story mode. The genes were Hammer Tail, Megaloceros Horns, Mammoth Foot, Armored Body, Digging Trunk and Saber Tooth Fangs. Happily, these can be discovered by mating with other creatures having these genes or - of course - by setting the genes on one of your packs creatures. The commands needed are:

• setgene pawmammut
  
• setgene hornmegaloceros
  
• setgene bodyarmadillo
  
• setgene tailhammer
  
• setgene diggingtrunk
  
• setgene sabertoothfangs
  
• Please note that a creature cannot have a Digging Trunk and Sabertooth Fangs simultaneously, as they both slot in the Snout gene.

Natural Selector
This is certainly the most time-consuming achievement, since you will have to unlock all genes in story mode. This can be done over the course of one long story mode session. Please note that you cannot just set genes on parents and breed them to unlock the genes. This only works with genes having the requirement 'Invite animals with this gene to your tribe' (e.g. the Bearyena and Prehistoric genes).

Darwin Award
Go extinct on the first island in story mode. Easiest way is by drowning Adam.