Information here comes from the wiki and my personal play experience.
Each Act has 3 potential bosses. Hexaghost is one of the possible bosses for Act 1. You can see if you're going to fight Hexaghost from the map. If there is a big hexagon at the top of the map, that means you will fight Hexaghost and can prepare accordingly. The good news is, this boss is pretty easy if you're prepared for it.
First, we should talk about getting to Hexaghost in the first place. If you're going specifically for this achievement, you can keep restarting until you see his symbol at the top of the Act 1 map. However, you will definitely face him many times while playing the game naturally. Most of the time, you'll want to prepare by improving your deck. In Act 1, this is fairly easy to do by fighting monsters and choosing cards (being careful to avoid losing too much HP). However, you can safely lose some HP since there is always a rest point before the boss fight. It can be difficult to devise a specific strategy while in the first act, since you are unlikely to have many rare cards or relics. So, try get a good balance of offensive and defensive cards. Buffs and debuffs are great too, especially for Hexaghost.
Once you are face to face with Hexaghost, you will see he has 250 HP. He attacks in a pattern, so I am going to list it below, along with the strategy at each step.
1. Activate - On the first turn of the fight, the boss does absolutely nothing. He just lights his floating fire orbs and cranks up the intimidating music. This is good news for you: you never need to worry about gaining block this turn, and can instead spend it giving yourself buffs or debuffing the boss. Another great option? Attack!
2. Divider - The attack deals H times 6 damage, where H is the boss' HP divides by 12, plus one. In other words, the more HP the boss has when he uses this, the more it hurts. "But it's turn 2, he's hardly taken any damage!" I hear ya. I suggest you block as much of this damage as you can, because he won't be hitting nearly as hard after.
3. Sear - Deals 6 damage and shuffles 1 Sear into your discard pile. Here's where the real trouble begins. Think of these Sear cards as your time limit. The boss will overwhelm you with these if you take too long. I suggest you just take the 6 damage and do more buffing/debuffing/damage here.
4. Tackle - Deals 5 damage 2 times. If you can, use 1 card that gives you block here to mitigate the damage and spend the rest of your energy on further buffs, debuffs, and damage. Hopefully by now you've finished using a lot of energy on status effects and can start focusing on dealing as much damage as possible.
5. Sear - Same as turn 3. At this point you have likely reshuffled your discard pile into your draw pile, meaning the Sear card from turn 3 might turn up in your hand. If this happens, feel free to use a block card, but otherwise just take the 6 damage and try to take as much HP off of this 6-sided jerk as you can.
6. Inflame - Hexaghost gains 2 strength and 12 block. This is another time limit factor, because that 2 strength means that Hexaghost will be permanently doing 2 more damage with each attack. That adds up quickly. You don't have to worry about damage this turn, so skip the block and really lay it on him. Ideally, he is now near half HP.
7. Tackle - Because of his +2 strength buff, this will now deal 7 damage 2 times. Hexaghost will also have 12 block on this turn, so don't even bother attacking unless you're going to be dealing at least twice that in damage. Instead, block his damage and apply further buffs and debuffs if you can.
8. Sear - Same as before, but now it deals 8 damage. As usual, I recommend you just take this damage to focus on doing damage of your own unless you have a Sear card in your hand or you're low on HP. If you are low on HP, don't "overblock" unless you have a defensive deck. Otherwise, you will lose the race to kill the boss.
9. Inferno - This will deal 4 damage 6 times and shuffle 3 burn cards into your discard pile. Then, it gets worse: all Burn cards get upgraded, meaning they do more damage. Now you're truly in a race against time. Block as much of this damage as you can, but focus from here on out on finishing him off before it's too late.
From here on out, Hexaghost will repeat steps 3-9, but with his strength buff (which will increase further) and upgraded burn cards (which you will draw more and more often). Normally the Burn cards deal 2 damage, but the upgraded ones deal 4 damage. Once you're at this stage, you're probably getting very close to death and you need to focus on damage, damage, damage. By the time he uses Inferno again, there is almost no hope whatsoever of survival.
So, to reiterate: with most decks, this fight is a race against time. After the first Divider, the damage output of Hexaghost is small, but he quickly powers up and fills your deck with Burn cards to overwhelm you. You need to kill him before he uses Inferno or as quickly as possible after. You won't last long past that point.
One final tip: cards and relics that can get rid of Burn cards in one way or another make this fight much easier. For example, the medical kit relic lets you play Burn cards to exhaust them. If you do this, you could theoretically survive all the way to the 2nd inferno without too much difficulty.