I personally think that the regular civilization of a certain barbarian village would take care of such things. While the warriors protecting them would only focus on their arms and strength.

My barbarian will be Dragonborn aswell. Should I start with the main quest or level up first?
I think a barbarian would be wise enough to understand that he needs to regain his strength. But a barbarian like mine would feel cowardly if he ignores the summons of the Greybeards and the dragon threatening to attack Riverwood.

Such hard decisions. Mine probably won't join the companions yet. Perhaps after the main quest.

The one that I played chased down deer and bludgeoned them with his warhammer.

I suppose the only ranged attack that I had was the shouts.

Food is underrated! At low levels it's actually pretty useful, just consider for instance Horker Stew (restores 15 health and 15 stamina + restores 1 point of health per second for 720 seconds). Besides, hunting and gathering your own ingredients and then cooking them is simply nice.
A barbarian would probably spend a lot of time in the wilds, pillaging and conquering others' dwellings. Maybe live in a cave somewhere... And dive recklessly into the MQ without further consideration.

He is the reckless type, so his way of hunting would be slaughtering a deer in cold blood with his greatsword. I thought about wandering from town to town, not exactly living anywhere. He would sleep at abandoned camps or camps of bandits he killed, most likely.

Would a true barbarian even care about helping someone he/she doesn't know? Maybe he would go after dragons if they attack him a lot, but not in defense of someone he doesn't know.

A pair of heavily reinforced gauntlets along with the Fists of Steel perk to allow you to beat people down would be rather cool. Really though, you've pretty much designed it all perfectly already.

His personality would be like one of the kind and sensitive barbarians, which is all because of his childhood. Those who grow up with a good morale.
Not to forget that Ralof helped him escape so he thought that a little help should not harm his image.

exactly, potions or magic don't fit with a barbarian build. If you're up for a challenge then do what I did, use only meat and mead to replenish vitals, only use a bow to hunt more meat and use no magic or enchantments. I wore no armor most of the game and only used a skyforge steel battleaxe but for your build fur armor seems suitable or maybe forsworn.
Remember to think barbarian in your actions for more immersion