1st level:
Rage could probably be activated as an immediate action, and would either confer advantage-esque roll twice for Fort saves and STR checks, or give a flat bonus, in the same way Rage does now. The damage bonus would remain the same as 5e - Add +2/+3/+4 to melee attacks that add STR (or 0.5x STR, 1.5x STR, etc.) to damage. Maybe it would get amped up a little, given that Pathfinder numbers are much bigger than 5e numbers. It might also add these to accuracy in a PF conversion, given that 5e is rather skimpy on accuracy bonuses (bounded accuracy), although it's possible to not do that.
Rage duration is mostly fine for a straight conversion, although 5e mechanics also pose a problem. Having up to 10 rounds of rage (per rage), so long as you make an attack or take damage, works fine in Pathfinder. However, in 5e, all martials effectively get pounce, and can take all their attacks with an attack action (although most martials only ended up with 2 attacks per round). Thus 5e martials can move and full attack every turn, and not sacrifice damage output, even if they kill a guy every turn, whereas PF martials would need enemies to line up to maximize the benefit of Rage. To do a straight conversion would lose that ability and thus reduce your damage considerably. To have a mechanically similar conversion would give pounce at 5th level. To keep balance, you'd probably need to compromise and give pounce at 10th or something, or cripple the bonus attacks until you hit 10th.
I'd be mildly concerned about interactions with Skalds and Raging Song (free rage), but perhaps it would be for the better. 5e Barbarians have a fixed number of rages (2-6), much like how Paladins have a fixed number of Smites. However, it takes time and class selection effort for Paladins to replenish their smites (2 Lay on Hands -> 1 Smite), and the trade is not so overwhelmingly favorable in the way Skald Rage -> 5e Rage is.
Resistance to bludgeoning, slashing, and piercing is fine. 1/2 damage is a thing that Shield Other does, so it would be fine for a Barbarian to have as well, especially considering that 5e Rage doesn't give bonus HP.
Unarmored Defense is a little tricky. Given the class interactions, you could easily stack 3 stats to AC (DEX, CON, WIS) unarmored, which are coincidentally also your save stats. If you're willing to go a little out of the way, you could end up with the alternative of (CON, WIS, CHA) unarmored. Or (CON, CHA), or (DEX, CON) in armor. Decent level investment would let you get (DEX, CHA, CON), (DEX, INT, CON), or (DEX, INT, CHA) in armor, or (DEX, CON, INT, WIS, CHA) unarmored, at high levels, although there are some alignment restrictions for the CHA to AC folk. But hey, at least you can't add STR to AC as well, right? The best move on this would probably be to level-cap it to a bonus AC of up to your level. It limits the Conan look to higher-level Barbarians, but it's probably worth it.
2nd:
Danger Sense could either use the Pathfinder Danger Sense (not a completely awful option), go for another 5e Advantage port and let you roll twice on (most) Reflex saves (pretty powerful), or go for some sort of fusion - give the Pathfinder Danger Sense bonus against events that would trigger the 5e Danger Sense.
Similarly, Reckless Attack could be ported over in this fashion, or could simply be a free Reckless Abandon. Reckless Abandon would probably be a better conversion, given the way Pathfinder works. If Reckless Abandon is chosen, it should be written out so that the class requires you to take the Rage Power, so that Extra Rage Power remains available.
3rd:
Primal Paths would need to be converted individually. I don't have a 5e book open right now, so I can't say the best way to convert each one.
4th:
Ability Score Improvement is covered in the Pathfinder system. I guess this is where you start choosing Rage Powers? You should probably be restricted from the 1/rage ones, and definitely from the ones that expend rounds of rage. Perhaps totems should be off-limits as well, given that the Primal Path abilities are basically totems. You'd need a custom Rage Power list, at the least. You'll also need to redistribute these to fill dead levels, if you're up for that.
5th:
Extra Attack is discussed a little earlier. If you want to start introducing a psuedo-pounce, it should either be here or 10th level.
Fast Movement is basically the Pathfinder ability, pushed back 4 levels. It's fine.
7th:
Feral Instict is fine for a straight port. Roll twice on initiative, and get a full surprise round if you rage.
9th:
Brutal Critical is a little underwhelming in the Pathfinder context. Rolling 1-3 extra dice on a critical is sort of underwhelming. A free Improved Critical, increasing the base critical threat range by 1 (up to 18-20 before Improved Critical), or increasing the critical multiplier by 1 would be a better way to go.
11th:
Relentless Rage would need a little touching up for the Pathfinder setting. The ability to return to 1 HP when reduced to 0 HP on a CON save in 5e basically means coming back from negative HP to 1 HP (or maybe 0?) on a Fort save in Pathfinder. The DC might also need adjusting, given that Pathfinder DCs scale. Perhaps a Fortitude DC 10+1/2 level+(5 per use) would be fair, since your Fortitude save is 1d20+2+1/2 level+CON+misc modifiers. "Long Rest" would be replaced by "when you regain your rage rounds", of course.
15th:
Persistent Rage is fine as is. Rage that stays until the end of your minute of power, without needing to bash people is great.
18th:
Indomitable Might might be a problem. I haven't run the numbers, but Pathfinder Strength checks are usually much smaller than Pathfinder skill checks, but your Strength score is expected to be in the 30s by this level. Your Strength bonus grows at 1/2 the rate, but will probably be something like +10 or +15, for an average of 20 to 25 on a roll. Not only that, but it scales twice as fast as a regular skill bonus from a stat. Granted, being able to do feats of massive strength is something Barbarian players like, so maybe it's worth keeping unaltered.
20th:
Primal Champion is fine. +4 STR and CON is basically what the regular Barbarian gets, except that regular Barbarians get +2 Will as well, but the bonuses only work while raging.
