Using a advanced strength potion which gives 260% Melee damage and a Diamond sword with a minor damage upgrade from the enchantment tables/books (I had sharpness II) on a monster will net you this achievement.
To make the Strength Potion II you will need.
2 x Blaze Rod
3 x Sand/Glass
3 x Cobblestone
1 x Nether Wart
1 x Glowstone Dust
1 x Crafting Table
1 x Furnace + Fuel
1 x Water Source
10 x Obsidian + Flint and Steel (if nether is required for materials)
Create a Brewing Stand from the 4th box in the Structure menu by combining 3 cobblestone and a blaze rod at a crafting bench. Blaze rods can be acquired in Fortresses from blazes in the Nether (get 2 rods to save time).
Use the 2nd blaze rod to create blaze powder from the Food tab either by pressing X or using a crafting table.
Use 3 sand in the furnace to create glass.
Use 3 glass to create glass bottles from the tools and weapon tab in the crafting table.
Find a water source block and fill the glass bottles with water.
Now return to your brewing stand. In the brewing stand menu.
Add the Water bottle to the brewing stand.
Add Nether Wart to a Water Bottle to create an awkward potion.
Add Blaze Powder to the awkward potion to create a strength potion.
Add Glowstone Dust to the Strength Potion to get a Strength Potion II
Drink the potion, once active red bubbles should emit from the player. Simply hit an enemy (I used a monster not sure if it works for animals) with the low enchanted sword for the achievement.
Further potions can be found here: http://minecraft.gamepedia.com/Brewing
If any steps are unclear, let me know I'll try and assist.