How do I get the "Overkill" achivements in Minecraft for Xbox One?

Overkill requires you to deal 9 hearts of damage, or 18 points of damage, in a single hit. For this answer I will be working in points (half-hearts).

A hit with an iron sword deals 7 damage. A crit multiplies this by 1.5 so it deals 10.5 damage. You need to get another 7.5 points of damage from other effects (enchantment damage isn't multiplied on crits). There are a few ways you could do this.

Sharpness adds 1.25 points of damage per level. Since smite doesn't normally go past rank V, this would only add a maximum of 6 damage, for a total of 16.5. On its own, this won't be enough.

Smite and Bane of Arthropods add 2.5 damage per rank against undead and spiders respectively. 3 ranks would be enough to crit for exactly 18 on an appropriate target. 5 would deal 19.5 damage on a regular hit.

Since Sharpness and Smite/Bane stack, you could combine the two at lower levels. Either Sharpness 4 & Smite/Bane 1 or Sharpness 2 & Smite/Bane 2 would work.

Your other option would be a Strength potion. A strength potion increases your damage by 130%- I can't tell how this stacks with the crit but even in the worse case (additive), you'd deal 7*2.8=19.6 points.

Just use a strength 2 potion on a mob

What you need to do for earning the achievement is dealing 9 or more hearts of damage to a mob, I would say wait until you have a diamond sword, as for enchants a would recommend Smite V for attacking undead mobs like Skeletons and Zombies or Bane of Arthropods for killing a Spider.

So enchant your sword and when you're going to attack a mob I recommend jumping and as you're falling strike the mob with your sword for a critical hit for extra damage.