To enhance equipment you will need either a Blackstone (Armor) or a Blackstone (Weapon).
There is multiple ways in getting blackstones. This includes from quests, buy them from the auction house, crafted using a refinery or from monster drops.
If you have completed the quest [Awakening] Traces of Edana from the Black Spirit, you will have gained access to Enhancement. You get this quest quite early in the game at the Ancient Stone Chamber, you will be around level 10 at this point.
While this achievement will come over time when upgrading your gear to +15, PRI, DUO, TRI etc. there is a quicker way in getting this achievement.
For armor you can upgrade it with 100% success rate to +5 and for weapons with a 100% success rate to +7.
So what you need for this is either 5 blackstones (Armor) or 7 blackstones (Weapon). Now go to Velia and talk to the blacksmith (Tranan Underfoe) or the arms dealer (Alfredo) depending on if you want to use the Armor or Weapon blackstones.
If you are only using 5 blackstones (Armor) go to the blacksmith and buy Reblath gloves, the gloves are 12,9k silver. You need 40 of them total which is 516k silver to get 200 successful enhancement attempts.
If you are only using 7 blackstones (Weapon) go to the arms dealer and buy Militia daggers, the daggers are 12,4k silver. You need 29 of them total which is 359,6k silver to get 200 successful enhancement attempts.
So what you want to do now is enhance the Reblath gloves to +5 or the Militia dagger to +7. Once the Reblath glove is at +5 or the Militia dagger is at +7 stop enhancing and go talk to either the blacksmith or arms dealer and go to the extract option. Put the Reblath glove +5 or the Militia dagger +7 in the extract menu and start extracting. It will extract all the stones you used (so either 5 armor or 7 weapon blackstones) and it destroys the piece of gear. Now just rinse and repeat this process untill you get the achievement.