for this achievement you will need to build a cauldron (7 iron). using a bucket (3 iron) carry some water from a nearby source and pour it into the cauldron. you will also need to make any color of dye. usually there are lots of flowers around, some of which can be made into dye. once made you can add this to the cauldron.
now you will need the leather armor to dye. a complete set (cap, tunic, pants, and boots) will require 24 pieces of leather. this can be acquired from large animals like cows, llamas, horses, etc. once you have the armor you can use it on the cauldron containing dye to change its color. [you can also use plain water to wash the dye back off the armor.]
NOTE: as soon as i dyed the leather cap, the achievement unlocked for me. this may be a glitch as the description says to dye all 4 pieces.