General answer: Half-Orcs
Looking at regular PH races exclusively, Half-Orcs are, from my experience, the most "suited" race. They often live in tribe-like communities, which fits to the way barbarians work in 5e roleplay-wise.
Also, stat-wise, they have initial strength and con bonuses - the most important stats for a barbarian -, they have "Relentless Endurance" (basically, you don't die when dropping to 0 hit points, 1/day), and they have savage attacks, so you deal extra damage on a crit.
Both of the latter abilities are also available to barbarians at higher levels - except they stack with a half-orcs abilities, so your barbarian gets even better at what he does.
Furthermore, proficiency in "intimidation" is pretty nice to have as a barbarian.
Other, less optimal, PH races
Other decent races include dwarves - you gain poison resistance and advantage on saving throws against poison, which increases your survivability, and you also increase your con and potentially strength stats.
Also, there's a dwarf-exclusive barbarian subclass, which could be interesting. I personally prefer the "standard" berserker, but to each their own.
Dragonborn also get a Strength increase, they get resistance to their respective dragon type, and they have a breath weapon, which can be potentially pretty useful (especially cone effects) since you're probably in melee anyway.
Aasimar get resistance to necrotic and radiant damage, and the 1/day flight ability of the protector subrace or the 1/day spheric radiant damage around the player of the Scourge subrace can be pretty nice as well. Note that the stat bonuses of this race are not optimal - the base race gives you +2 charisma, which is (for you) mainly useful for intimidation, which you might often be allowed to do as a strength-based check. Protectors get +1 WIS, Scourge Aasimar get +1 CON (which is useful), and Fallen Aasimar get +1 STR, obviously also helpful.
Hence, overall, they are only a mediocre race for barbarians.
I will add details for other non-PH races later (i.e. Volo's Guide, where Aasimar are from), since I currently don't have any rulebooks on me and recalled this information from memory.
I'm guessing the "Orc" playable race from Volo's Guide will be pretty nice as well, but I haven't really taken a look at that yet, so I don't know without a rulebook in front of me.
There we go:
Earth Genasi (Elemental Evil): +1 STR, +2 CON; they don't have any actual abilities that are of great use to barbarians, so not really the best race.
Goliath (EE or Volo): Easily one of the best races for barbarians aside from Half-Orcs; they get +2 STR and +1 CON, gain Athletics proficiency, can shrug off 1d12+CON damage once per short or long rest, and a few minor other abilities.
Bugbears (Volo): +2 STR and +1 DEX, +5ft weapon range on attacks during your turn, Darkvision, stealthy and their own version of sneak attack once/combat.
Orcs (Volo): Surprisingly less efficient than Half-Orcs. They gain +2 STR and +1 CON, can move up to their speed towards an enemy as a bonus action, gain Intimidation proficiency and darkvision.
If you're going to go for the Totem Warrior subclass and choose the bear totem at level 3, ignore anything mentioning resistance in this answer. Since you will get resistance to all damage types except for psychic, racial resistances get pretty pointless. The cases where you will take damage while outside of a rage are pretty rare, and it's not very likely that the few resistances you get from your race will be relevant in that case.