Before you start crafting anything, you should collect the following:
-4 lapis lazuli
-2 ink sac
-1 cocoa beans
-16 white wool
-3 oak planks (or other fuel for smelting)
NOTE: If you come across a gray, light gray, brown or black sheep you can use their wool for the achievement instead of crafting it yourself.
Once you've collected all the necessary ingredients, follow these steps:
1. Smelt the cactus blocks in a furnace to get 3 "cactus green."
2. Save your game, just in case!
3. Make 2 yellow dyes using the dandelions.
4. Make 4 red dyes using the poppies.
5. Make 6 bone meals using the bones.
6. Make pink dye.
7. Make orange dye.
8. Make lime dye.
9. Make gray dye.
10. Make light gray dye using the FIRST option (gray dye + bone meal).
11. Make light blue dye.
12. Make cyan dye.
13. Make purple dye.
14. Make magenta dye using the FIRST option (purple dye + pink dye).
15. Now comes the fun part. Scroll over to the wool column and, starting with orange wool, go down the list crafting one of each wool until you reach white. If you did everything right, you now have one of each color with some dyes left over! Awesome!
There is a common glitch where the achievement doesn't pop when it should. If this happens to you, the best thing to do is throw all the wool blocks out of your inventory and pick them all back up at once.