Base Class: Barbarian
A reckless, death-defying, thrill-seeking daredevil with a font of courage that can't be corked.
Heart of Fire
Whenever you take a reckless attack action while raging, you gain an additional 1d6 bonus damage die when you hit.
At higher levels: Your die increases as you take more levels in this class. You gain a 1d8 bonus die at lvl 6, a 1d10 at lvl 12, 2d6 at lvl 17, and a 2d8 at lvl 20
The Harder They Fall
Once per day, you can choose to ignore the damage from any one source. For every 5 damage that this ability negates, you gain an additional 1 damage bonus to your next attack.
Bite the Dust
When you kill a creature with a melee weapon attack, you gain temporary hit points equal to your level + the bludgeoning, piercing, or slashing damage you dealt.
Total Meyhem
When you hit a creature with a melee attack, you may apply the effects of the spell confusion without its spell components to that creature. It’s save DC for this effect is equal to 8 + your Strength Modifier + your Proficiency Bonus.
damage at 6th level go brr
small note include that you can't use The Harder They Fall against crits
You forgot to mention the level in which they get each feature.
I like it, but I will also say that it is quite strong. It can literally rival Totem Warrior in terms of how powerful this subclass is. Though, considering that Totem Warrior is actually in the game, this is another pick that people will chose due to it's immense power. One complaint I have is how powerful Bite The Dust is. as Kleinykun said, I also think that barbarian level instead of damage dealt would be better.
bite the dust should only be barbarian level. barbs are already the most tanky class they don't need to be anymore tanky. also the harder they fall needs tweaking. not much.
Heart of fire is absurd. barbs already dish out a crap ton of damage they don't need even more. maybe the first attack that hits deals and extra 1d6 (since path of the zealot also has that)
Total mayhem should be 8 + con mod + proficiency bonus. also should only be a number of uses equal to twice your con modifier and before you know the outcome of your die roll.