Dump strength. I know what you're thinking. BLASPHEMY.
As soon as you get into the game, set yourself upon finding a belt of giant strength, and then when you find it, obsess over finding a better one.
Problems solved.
To do this, just find and attack the biggest things you can find looking for strength. The belts are all from giants - take giant as a language. Ogres, namely Oni's, would be especially adept at procuring vital parts for such an item. Be relentless.
To aid in this, be weak. Be feeble. Use Cower and Beg from the kobold list. Make it "a thing". BE the kobold barbarian whose dream is to be strong - whose scurrying, feeble race can't pickup greatswords, that halberds are too big for. This will be hard, because you will be weak, but play out that frustration. BE FRUSTRATED. RAGE. Sooner or later your DM will cave, or you will have just done so much to get that damned belt you can just have one enchanted for you. It'll be a long, slow trek. BUT IT'LL BE WORTH IT.
Until then, suffer under using Dexterity VS Strength. Focus on Finesse weapons. Whips, Shortswords, Scimitar's, Rapier's, Daggers and Darts. Bring plenty of Daggers (DM's will call you on being a small creature with 15 javelins on you) because you need to make an attack to keep your rage going. It doesn't distinguish it need be a melee or strength attack, so daggers will do just fine.
You will lose your Rage damage, but negotiating with your DM to get this on Dexterity based checks isn't a hard sell. It'll require training in game - probably with monks to fuel rage into dexterity, then flavor it as you balancing your rage with your acrobatics. It's definitely the biggest loss, because you lose that +2 to +4 (equal to a 1d4 or 1d8 extra damage). Your rage will become, mainly, a means of getting resistances - for pure defense. It'll also give you advantage on strength checks, which will offset how feeble weak you are.
Shield Master, Dual Wielding, Sentinel are great choices. Defensive Duelist will make you a bruiser in a melee fight.
Your AC will be quite high if you punch Dex and Con, due to unarmored defense, and you will block like a boss even at level 1.
Now, the belt of Giant Strength will ultimately render your level 20 / level 18 barbarian abilities largely moot. To fix this, I would recommend multiclassing 1-2 levels in another class to shore up your skills. You can wait until the very end if you like, but you can squeeze 3 levels in and only lose 1 ability score (which you won't need), indomitable might and primal champion. Losing Primal Champion sucks, but defensive duelist makes it moot as you can pump your AC anyway. The strength bust won't matter, because your belt does that anyway if you keep improving it. Indomitable might (guaranteed 19 on a str roll) is great, but it's more fun to roleplay that little kobold occasionally getting suckered even WITH the belt on.
Instead, I would plan on multiclassing. Into what, and when, is up to you, but those last 3 levels of barbarian are pretty useless to you. Multiclassing early pushes your multiattack away, and starts messing with game balance - it might be best to wait until, say 11 in barbarian. But at some point you'll want to multiclass. Rogue seems an excellent choice, but I'd stay away - eventually you will have to decide between sneak attack and STR with the belt - and when you get that STR, you'll want to go back to STR weapons.
Fighter for 3 isn't bad, especially if you go Champion to get crits on 19. Given your reckless attack and brutal critical, you will be a crit-fishing champ. Literally. Action Surge + Second + Fighting Style will go a long way. Ranger's get Absorb Elements / Zephyr Strike which are great and their basic class abilities are a nice touch. Ranger, paladin, fighter.... these do all a little bit of overlap for you though, as the proficiencies you get some you already have, or will never use.
Sorcerer / Bard / Wizard could give you a lot of utility. You will have focused Con / Dex, but just choose buffs that don't rely on your spell save DC. Sorcerer subclasses don't have much for you, which is a pity, because their features are great. After getting your Belt suddenly dipping 3 levels of bard might calm your DM down when he looks at your ability scores.
Clerics are fun. Warding Bond on a barbarian is siiiick since the resistance is recalculated after you take damage. It takes a while to figure it all out, but barbarian resistances do work on damage transferred by warding bond (since all damage has a type) and it'll keep you protecting somebody, though AOE's are bad news for you (150% damage anyone?). Shield of Faith + Sanctuary have staggering implications for making you super annoying to deal with.
Warlock's are always a great dip, and Eldritch Blast is a ranged attack to keep your rage going - and you can do a blade pact. Gift of the Ever Loving Ones (max healing!) is a great one if you're playing with a cleric - but also can be used via potions.
Druid's are excellent - circle of land gives you access to a bunch of unusual spells, and wild shape has great utility even when you can only do minor creatures.
But regardless - you know the strat. GET THAT BELT.
