Did Captain Barbossa continue his career as a privateer in the fifth film?

No, he's no longer a privateer, but a powerful and wealthy pirate controlling the seven seas.

At one point towards the end of On Stranger Tides Barbossa declares,

"I care not for King George or tavern yarns that give hope for a healed limb. But I'd give my left arm for a chance at Blackbeard."

and later after Blackbeard is defeated,

"All hands! Ply to windward! Get cracking, ye bloomin' cockroaches! [laughs] The Crown served me well. But now, by the Gods of sea and sky! Make way for Tortuga!"

Hector Barbossa returned to a life of piracy and at the beginning of Dead Men Tell No Tales, he's become 'the terror of seven sees' by use of Blackbeard's former sword and Queen Anne's Revenge...

He became the terror of the Seven Seas, pillaging the wealthy trade ships and filling the holds of the Queen Anne's Revenge with treasures beyond most men's wildest dreams. With the power of the Sword of Triton, Barbossa finally fulfilled his old desire of living the life of a rich rogue. He surrounded himself with riches, dressing himself in finest uniforms he could steal, and even replaced his wooden peg leg with one made of gold. He successfully took over the pirate world, his name being whispered in fear across the Caribbean, and when his Jolly Roger was spotted at sea, pirates and soldiers alike knew it was only a matter of time before they were conquered.

It also becomes clear that there aren't many real challenges left and that he almost seems bored with this life style and it's the challenge of Salazar and reveal of Carina that seems to give the character some real meaning to his life back.