This achievement is no joke. The Wither will break any block so you can't trap it (there are guides online saying you can trap the Wither, but when I tried they did not work).
I tried to follow a specific guide about trapping the Wither in the ceiling of the Nether. It did not work for me, so I do not recommend it. What happened was the Wither was not trapped and destroyed the floor under me so it became impossible to fight.
What I did to eventually kill the Wither was on the normal overworld dig all the way down to the bedrock. Once down there make sure you have an easy way to get back down because you will die. I also highly highly recommend that you do not do this by yourself. You need someone who can hide and keep track of the Witcher's health.
Once you spawn the Wither he is in ranged mode. He will fly above you and shoot skulls at you. You can knock the blue skulls back but arrows are easier. He wants to kill you so he won't fly away, but he will stay out of sword range. You will need a bow with Power 5 and Infinity so you can shoot him down. You will also need potions of Night Vision so you can find him. Any torches you put down he will destroy so just use night vision. Once you get him to half HP the fight changes.
He summons 3 wither skeletons and then goes into melee mode. In melee mode he shoots 3 skulls at you and then dashes towards you breaking all of the blocks he goes through. He will not heal past half HP so you don't have to worry when he kills you. At this point you need your strongest diamond sword with Smite 5 or Sharpness 5. The easiest way to deal damage is to get under him. He can't shoot skulls straight down so you will be able to get a lot of good hits on him.
I don't recommend wasting time building armor. It will let you stay alive longer, but with the Wither effect he puts on you and the damage he does he is going to kill you.
The reason you need a second person to watch his health is because if you go out of range on the encounter he will heal. If you have someone in the area and can see the HP bar he will not heal. You have to be careful because if the person is too close the Wither will find and kill them. I had a friend burrow into the ground near the Wither and watch his health as I died and came back. It was the only way I won.
To set up I would recommend having a box full of bows, swords and night vision potions near where you will fight. Do not have it too close because if the Wither fight goes to the box he will destroy the box and you will lose all the contents. When I did the fight I had a very long tunnel between my base and the Wither (I am talking like 300 blocks) I had the box at the beginning of the tunnel so I would grab what I needed before I ran back to the fight.
This is not an easy fight and it's hard to prepare for. Good luck.