For the current tutorial I have collated information in the comments, I believe this may be the last iteration on the X1, I have also left up the previous methods if they are applicable.
You can now load up the tutorial, dismiss the original tutorial messages and then head straight forward, head past the beacon towards the sign with the wings on it, below this is a chest with 10 leather in it, grab that, shut the chest and leave the immediate area, then head back and the leather should have respawned, so grab it again, this will give you 20 pieces of leather in total...we need 24, so repeat the trick and you should have 30 pieces.
Thanks to
xPut Name Herex.
With the latest update, chests respawn the materials in them when you re-enter a tutorial area. So you can go and grab those leather in the wings area, turn around and leave, then come back and grab the batch of leather again until you have enough. Just make sure to dismiss or complete the "do you want to learn how to do this?" prompt, as the game won't refresh the chest contents when it thinks it is mid-tutorial already.
Head back towards the beacon and stop at the chest with the banners and in there should be your Dye, pick any colour you wish. Then turn right and head down the steps where you should find the crafting table. Craft the four pieces or Leather Armour required, Helmet, Shirt, Leggings and Boots. Don't wear any of the pieces, keep them in your inventory.
The recent updates have added a menu for dying, explained below by
OptimalShadezThe seems now that in order to dye your leather armor you need to open your crafting menu. Press LT and it give you a new menu to dye armor.
One final piece of advice from
SpeedBumpVDrop is
I'd like to point out that dying it with the Cauldron won't work.
Hope this helps, let me know if any further information would be useful.
TU 46 Method
*** Spoiler - click to reveal ***This is for the new tutorial level launched after TU 46 I believe:
I will improve this part of the solution with coordinates etc. once I have access to a computer.
From the beginning of the Tutorial head straight up the stairs and turn left at the top to the animal tutorial section. Kill the cows to get at least one piece of leather.
Head back the way you came and then turn left to face the beacon, go up the stairs either side of the beacon and keep heading straight till you get to the banner tutorial section, open the chest and grab any dye you like.
Keep heading in the same direction and you'll get to the new wings tutorial sector...in this area are 10 pieces of leather, grab these and head back through the banner section, turning right shortly after to get to the armor area. Here use the crafting table to make leather boots and leggings.
Now for a short journey, head directly West across the water and mine cart tracks, then go either over or through the mountain ahead...you will see a wooden bridge, cross it, turn left and cross the next bridge. On your left is a door in the Tower. Make your way to the top of the tower and you'll find a chest with a leather helmet and chestplate.
Finally press 
to enter the inventory page, grab the dye and hover over each item of armour...pressing 
to dye each one.
There have been some issues with the achievement not being tracked, please find below two possible solutions offered by fellow sufferers:
Barrwg offers this
I made my xbox go offline then started up Minecraft on a new tutorial world. I grabbed 10 dye and 30 leather (using the whole resetting items feature when you "redo" a tutorial) and made each leather piece and then dyed them.
After that I saved and quit, closed minecraft (start button, quit on dashboard) and reconnected. Booted the game up and my xbox started to resync the saves. Went back on the save and redyed each of the pieces with the same colour and the achievement unlocked after about a minute.
Not sure whether the resyncing fixed it or what, but might be worth a try.
and https://www.trueachievements.com/Game+of+Jawns.htm suggests this
Created a new world, named it helmet, unchecked online play, peaceful.
Collected a flower and 8 hides, set up a crafting table, turned off auto save, then made a new save.
Crafted a helmet, dyed it, waiting for the loading icon in the upper right to vanish, then exited without saving.
Loaded up that world again, crafted a chest plate, dyed it, waiting for icon to disappear, then exited without saving.
Repeated steps for pants and it finally popped.
TU 22 Method
*** Spoiler - click to reveal ***Below is for Title Update 22 Tutorial:
As stated by the other solution, the way to dye leather armour is to press to enter the equip armour menu, pick up a dye and hover over a piece of armour, then press to dye that piece. The helmet, chestplate, leggings and boots all have to be a different colour dyed.
For the more fashion conscious among you Mike Marcelais added:
You can use the same color for all four pieces. (I used Dandelion Yellow for all of them.)
The hard part is finding the armour or leather to manufacture it. What I found best was to load up the tutorial, then head to the beacon in the village, (follow the beam of light).
On the opposite side of the street is an enchanting room, if you face this room then enter the building to the right, follow the left wall past three dyed wool blocks and head out the back door. In this garden are enough plants to get 4 different colour dyes...and in the distance is a large church. Inside is a villager that trades leather leggings and an enchanted leather chestplate. (If you used the duplication glitch in the solution for beaconator then you will have plenty of leftover emeralds to trade).
This leaves you with only 9 leather pieces required, 4 for the boots and 5 for the helmet. I just ran around killing cows and horses because the tutorial has far too many animals already active to make breeding a viable option.
Slight amendment from https://www.trueachievements.com/CanadianWhizKid.htm:
Doesn't have to be all at once either. I made one piece and dyed it. Then I exited without saving, reloaded, and made the next piece of armor, doing the same thing with each piece.
I got mine in about 15 minutes, head out of the church and turn right, skirting the village back towards the start...I found a couple here, then head to the breeding section and kill those, then head round the back of that cow pen and you'll see a village in the distance, I found 6 on the way to this village. Unfortunately, other than the breeding pens I don't think any of these are guaranteed, but I wish you luck.
Another place suggested by YAT0:
A great time place for me to farm cows and horses was tutorial x 201 y 68 z-190. Keep running around and killing the animals you see. They seem to respawn in this area quickly.