I thought so aswell. But I really like to stick with my story. And I don't see somebody with alchemy knowledge in a Barbarian.
I agree, I'd suggest perhaps a little smithing, repairing or improving fur armor, etc., might fit with your character concept.
Try using melee weapons exclusively, no ranged weapons, no spells, no sneaking around and just run headfirst into battle. Recently played a 2-handed barbarian type character who considered those to be cowardly activities. It was refreshing change after having played stealth and mage characters previously. For this character, I put all my perk points into two-handed weapons and stamina, no health or any other skills.
One warning, though - I've played a few different characters with different playing styles and Bleak Falls Barrow was insanely challenging for this two-handed melee character at level 1 on Master difficulty with no mods. I died dozens and dozens of times in the room with two restless draugr. I couldn't stagger them to interrupt their spells because they were ganging up on me, and I didn't have any ranged attack at all and very little stamina. From that point forward the game became easier, and the Bleak Falls boss was not too hard compared to these guys.