Credit to ZSlyzer for the first video to this solution.
Credit to bubufubu for the alternate video to this solution.
Obtaining this achievement is a piece of cake, since you encounter this team very early on in the game. Try to have the axe on you, which you can obtain some rooms prior to this one.
This duo works together, with Gaibon carrying Slogra around. Slogra usually thrusts with his spear, but sometimes will dash at you, "nose" first in a thrust move that also causes damage as well.
Gaibon flies around, traditional gargoyle style, and tries to breathe fire on you.
However, if you follow the video, you'll avoid even seeing their attacks really.
Here's a description: When you attack Slogra, you want to use a quick weapon like a knife since you can spam the attack button. When Slogra takes damage, he'll jump up into Gaibon's grasp to be carried around, where he attempts to dive down on you from above. To avoid this, simply head to the right of this room, and you'll see a bit of the wall hang down at the same height these two are hovering. If you stand just below that wall, they'll attempt to continue flying right, but with the wall blocking, they'll be stationary. Every time Slogra drops down, you'll be within range to attack him and force him back up almost immediately (since you're standing right next to his drop down location with a knife). Mix in some Axe usage as well (the axe being the special weapon that uses hearts) for when they're in the air, so that you can continue to pummel them with damage. Eventually, Slogra dies (bursting into flames) and Gaibon dives down (slowly) to fight you on the ground. Simply spam that knife at him and he'll soon fall (especially if you worked some Axe hits on him earlier).
Like I said, fairly easy for an early-on boss battle.
If you want another strategy, check out the second video where bubufubu uses a basilard, the axe, and the tetra spirit magic. To perform tetra spirit, charge UP for 2 seconds, then perform a half clockwise circle down (Up, Up-Forward, Forward, Down-Forward, Down) and X upon reaching "down".
His stats were:
STR - 8
ATT - 9
INT - 10
Which is common for this point in the game.
Use whichever strategy you prefer for encountering these two (or make up your own!) and you'll have this achievement in no time.