Dude, you're referencing a portion of a spell that fluffs changing the damage type and amount of your Unarmed Strike as growing claws and stuff. A Bear doesn't even have Natural Weapons, it has Claws. They are even labelled as such.
So sure, you can use Natural Weapons as Unarmed Strikes! Too bad the only times the game references natural weapons, it does so as fluff, not as a rules term.
Like you said, 3.x =/= 5e; in 3.x, Natural Weapons were a rule term. In 5e? They aren't.
It's like if I wrote a spell that had the FX of setting you on fire, and said that it let you use the Dodge action as a Bonus action; it would be really silly if you looked at that and went "whelp, being set on fire lets you take Dodge actions as a Bonus action."
Hunter's Mark is only on the Ranger's list (and you can't get it with either dipping Ranger or 6/10 levels of Bard)... so how are you casting it? Also, are you assuming you have it pre-cast on the guy you are slaughtering, or are you taking into account that you need to spend an Action to fire it off?
You also need to reduce the damage dealt a bit to account for AC (not by much, though; you do have +15 to hit and Advantage, after all (unless Oathbreaker boosts attack rolls too).)
Other than that, it looks pretty fine for "spike" damage.
Funnily enough, though, Rogue and Barbarian is a nice combination, even if you aren't trying to break the game. After all, you can use Relentless Attack to get a Sneak Attack off every round, regardless of whether or not you have anyone backing you up.
Also, you can get some good jumps with a good Strength, Expertise, and Rage.
TenaciousJ, Sneak Attack is 1/Round in 5e.