Literally any character class in 5e would work.

Barbarian - His tribe was torn asunder by dark magic, and he has been filled with raging fury to track down and punish all those who use it, all the while following the trail of his tribe's original witch doctor, who began it all.

Bard - You use your disguise and performance skills to infiltrate the highest echelons of society, hunting cultists and demonics who practice in secret, protected by riches.

Cleric - A priest who worships the God of the Hunt. Initially a simple bow hunter, he grew bored with his quarry, and feels compelled to hunt more dangerous game ...

Druid - A moon druid, who is studying (at early levels) to harness the power of the elemental plane, and has a secret desire to one day learn how to shapeshift into a demon, so he hunts and tracks them.

Fighter - Very broad. Any background really, someone who picked up a weapon and thought "this would look awesome plunged hilt deep into a Tanari's stomach!"

Monk - From a secret monastery, high in the mountains. You appeared on their doorstep one day as a child with no memory. They sheltered you there for ten years as you learnt their ways. One day a demon came and destroyed the entire sect. Only you escaped. You hunt demons, but they also hunt you ...

Paladin - Oath of Vengeance to hunt and slay demons. The Oath is something you are committed to, but primarily for the powers it grants you to track demons (sense evil, etc) rather than any moral code. Paladins in 5e aren't restricted to LG.

Ranger - Well you said you didn't want this one but a Hunter ranger could still be very powerful.

Rogue - You've always been drawn to darkness. You feel comfortable there, and nothing brings you a greater rush than hunting those who dwell in it. Ever the thrill seeker, you find yourself tracking deadlier and deadlier prey.

Sorcerer - Work with your DM to come up with a divine / celestial sorcerer origin. Your blood burns when demons are near, and you are filled with a near-insatiable rage to find them and punish them.

Warlock - Perhaps with a fiend patron who is seeking to manipulate power structures in their plane of hell, and kill their rivals. Or perhaps The Undying Light!

Wizard - Ever seeking knowledge, the magic of demons is known to be filled with near limitless power. You want this for yourself, and will do anything to get it!

Finally, perhaps one of the most apt choices (though it's homebrew), you could try Matt Mercer's Blood Hunter.