Here are all the Barbarian subclasses broken down and ranked in the video, but I tried to make character sheets for each one, while optimizing them the best I could, and adding a brief description as to how and why I did that for each.
I should note, all the Barbarians have the same background for optimization purposes. They all have Wildspacer to make sure to get the Tough Feat right away for that bonus health.
Berserker: https://www.dndbeyond.com/characters/124311411/Xi03RY
Berserkers downside of having to deal with exhaustion was the main focus, so I went with a Wood Elf for the race, which really helped in a lot of ways. Once you hit 2 levels of exhaustion, your speed is halved, but Wood Elves are a bit faster than other races to help mitigate some of that even if your Barbarian Fast Movement helps more. But the big help comes from their Trance ability, being an Elf you can take a long rest in 4 hours, so if your party needs a long rest, you can take 2 long rests getting rid of 2 levels of exhaustion at once. Then the final benefits are that we can have Elven Accuracy, basically gaining double advantage. With regular access to advantage thanks to Reckless Attack, you have a drastically increased chance to crit, so I also took Great Weapon Master giving you a better chance to get another attack as your bonus action without having to rely on Frenzy. So in the end you can have Frenzy more as a last resort, but not be worried about using it thanks to Trance.
Totem Warrior: https://www.dndbeyond.com/characters/124314467/Dvx8vA
Bear is one of the most common types of Totem Warriors for a reason, they're good. I focused on that tanking even more by outfitting a shield, and the Goliath race. By being a Goliath you have the option to use Stone's Endurance, using your reaction to reduce damage by a d12, and since you're already resistant to almost all damage, you become incredibly durable. Another option was Githyanki with their Psychic resistance, which would officially make you resist everything, but that's a far less common damage type, so Goliath seemed better.
Battlerager: https://www.dndbeyond.com/characters/124314683/81Xlc9
This is one of the worst barbarians but to try to increase the Battleragers core abilities, I focused a bit more on the grappling. Battlerager spiked armor gives a bonus 1d4+Strength of damage on a grapple, then I grabbed Skill Expert to make sure you never fail a grapple, and Fighting Initiate Unarmed Fighting for another 1d4 of grapple damage. Then I topped it all off with the race Grung, while they do have the big downside of Water Dependency, that's easily taken care of with someone casting Create Water. The big bonus from Grung is Poisonous Skin, just being in contact with your skin can poison them, and with all the grappling, that will happen a lot. Added a War Pick for the weapon for one more way to apply poison, since it requires a piercing weapon.
Zealot Barbarian: https://www.dndbeyond.com/characters/124314967/wF6g8S
Going Protector Aasimar really boosts the Radiant Damage you can deal as a Zealot, and if your Rage Beyond Death is about to run out you can use healing hands on yourself, becoming basically invincible.
Beast Barbarian: https://www.dndbeyond.com/characters/124489338/SBluUi
Beast barbarians get a decent variety of options already, so I went with Leonin for the race, it leans more into the Beast part of roleplaying. Also their Daunting Roar giving you the ability to frighten nearby creatures uses your constitution modifier which gets super high as a Barbarian. Also I gave a climbing speed for the Beast option, because that tends to be the most useful.
Ancestral Guardian: https://www.dndbeyond.com/characters/124489634/470Z4l
Outfitting the Ancestral Guardian with a Shield and Warhammer emphasizes the tank role boosting their Armor Class. Amping up this idea by going with a Beast Hide Shifter helps, giving more AC and some temp HP when shifted.
Wild Magic: https://www.dndbeyond.com/characters/124489916/SOU9yD
No perfect way to optimize this other than stats, and being a Protector Aasimar gives the opportunity for a nice boost to damage and flight.
Giant Barbarian: https://www.dndbeyond.com/characters/124490302/cHsS4v
This one may have been rated a bit lower than it should in my video, but people have made a solid case for why it's so good. The best simple way to improve this for optimization is going with a Scourge Aasimar, their Radiant Consumption does damage in an AOE Aura. That AOE damage becomes massive in its range as your size grows being a Giant Barbarian.
Storm Herald: https://www.dndbeyond.com/characters/124490472/XwbhO9
With the Storm Herald I chose Sea Storm, allowing for some additional lightning damage without worrying about damaging allies, and simultaneously giving some chances to knock your enemies prone. And to still get some AOE damage I went with Scourge Aasimar. No really great ways to heavily optimize this one.