In order to obtain this achievement you are going to want to start off by creating a new world with the following settings:Default Game Mode:
1204739880 (or just click the arrow button on the right and select Savannah Village from the Seed Picker option)
This will start you off in a world with a village very close and a river running near by. Your first task will be to get a handful of raw fish by simply jumping into the river and looking for fish swimming around. You are going to want to melee each fish 3 times in order to get them to drop what you need. Once you have around 15-20 pieces of raw fish I would then suggest you travel to the village located nearby and create a save and then quit the game.
What you are going to want to do is manipulate the save by making a duplicate of it and then using that duplicate file to feed the raw fish you have obtained to the stray cats in the village. In order to do this, just select the little pencil/pen icon on the right side of the world's save file and scroll all the way to the bottom to the option that says, 'Copy World'. This will create a duplicate save.
When the above has been done, simply load up the duplicate save. When you have spawned, make sure you have the raw fish you have collected selected and try to locate a nearby cat and crouch nearby it using
. Do not push up too close on the cats at they will get scared and flee. When a cat gets close enough, aim at it and press the
button to feed it the raw fish (the cat will have love hearts when it has been tamed and may take multiple fish to do so). Do this for all the cats on that save within the village and then save and quit.
Rinse and repeat the above process by copying the main save and creating a duplicate file until you have fed a total of 20 stray cats. This method speeds up the process without needing to farm raw fish each time. From scratch, this should take approximately 30 mins. PLEASE NOTE:
My achievement tracker did not go up after taming each cat, but did give the achievement as soon as I had tamed 20 cats.