Just to be clear; the character wants to punish/kill/take vengeance upon slavers and orcs? Don't mean to be dense, but I couldn't quite tell from the thread title if he was a slaver hunting orcs or someone hunting both that profession and that race. I'm going to assume you meant he is anti-slavery.
Alignment questions are going to generate a whole slew of opinions on this site. Aside from a few very specific spell and class feature rules, alignment is very hard to nail down. The "rules" are almost all in prose and descriptive language, as you can't really put numbers on abstract concepts like morality, good, evil, law, chaos, etc. Despite what some will so passionately and elegantly argue, you can't put hard and fast rules on these things. Alignment is at best a character concept tool; a guidepost to direct ways in which your character behaves. The problem is, we all have our own cultural, religious, political, and personal views of morality; and we can't help but bring those to our perceptions of the game, even when we fully understand it's all fantasy. There are just too many variables, and too much entanglement.
All that said. Let your campaign and group guide you. If slavery is legal within the culture or realm of your characters background, then he's neutral or chaotic. If slavery is a no go, then his reasons for objecting to the act come into play. Does he oject because of the cultural/legal taboo, or only because of his personal experience. Would he have cared if it was just a story he heard about some farmers three valleys over? One indicates for a lawful character, the other goes to neutral.
Good/Evil gets more finicky. Both his motivations, and how far he's willing to go factor into this decision. If he's vengeance at all costs and innocents be damned, neutral at best, slipping towards evil if he's willing to be a bigger monster. If there's some moral outrage behind his mission, a whole "never let it happen to anyone else" attitude, maybe he's good.
Class wise you've described a few different features. The flying into fury thing sounds straight up Barbarian, especially if he has little background in formal martial skills. On the other hand, if he's more disciplined, or has better training; you've got a good set up for a Ranger with favored enemies of orcs, humanoids, and maybe giantkin.
Paladin works if you're the all around good guy with law on your side.
But be ware, Paladin plus alignment questions is going to open this thread to a whole lot of crap.
Sounds like it should be a fun character. Maybe a bit grim, but some good roleplaying opportunities. Good luck.
