Your first character can use all kinds of crazy options, including things from other books like Xanathar’s Guide to Everything, or homebrew found on the Internet like the one you link. If that’s what you want to do, don’t let anyone (well, other than your DM) tell you otherwise. My first character was a sorcerer with all kinds of things going on, and even from the beginning I was asking my DM if he’d allow this thing or that—so it can be done. I loved reading up on all the different options and thinking about how they could work together and what kind of character I could make.
But often, new players find it easier to stick with more basic options. I did a lot of reading and research to make that first sorcerer. I even started a forum thread (not here) that went 127 posts long, discussing my options, before I’d even officially joined a game. That was a lot of work—work I loved, but a lot of work. If you aren’t interested in all that, this answer might suit you better.
And that answer is to stick with basic options, and allow your imagination fill in the blanks for your character, rather than needing some book to tell you that you’re playing the character you want.
It’s worth noting here, I think, that page 140 of Player’s Handbook—where the soldier background is described—has a picture of a woman in traditional samurai armor, holding a katana. This is despite the fact that neither “katana” nor “samurai” appears anywhere in the book. That’s because being a “samurai” is about more than picking a class or background or anything else that has “samurai” in the title. Even if you did have a class that was called “samurai,” that wouldn’t necessarily make your character one, and just because you don’t have anything called “samurai” doesn’t mean you aren’t one. This fallacy is a trap that a lot of D&D players fall into—it’s so common that Order of the Stick made a joke about a samurai with no levels in samurai classes—, but it’s really important to recognize that it is false. Your character is a samurai because of the training they received, the code of honor they adhere to, the outlook they have on life—not because of what their character sheet says. The character sheet can only say so much, after all.
So with that said, if we ignore the labels on things, how could we create a character like Jack?
Samurai Jack fights with a katana; those are actually really heavy weapons. And he spends most of the show wearing nothing more than a tunic, rather than heavy armor. So if you want to play a character like Jack, you have to simultaneously wield a big weapon, and wear no armor. Do any classes fit that mold?
Fighters certainly wield big weapons, but they’re usually seen in armor, too. And monks famously fight unarmored, but they also usually fight unarmed or with “monk weapons,” which would not included a heavy two-hander like a katana. You could use either of these if you really wanted—nothing says a fighter has to wear armor, or that a monk can’t use a heavy weapon—but it seems like kind of a waste.
But the barbarian class gets unarmored defense, just like the monk. And barbarians certainly use heavy weapons. But is Jack a barbarian? He’s definitely not barbarous, he’s actually fairly cultured and refined. But he also has anger issues, and certainly has a kind of relentlessness and stubbornness that barbarian seems well suited for. So you could kind of see Jack as a barbarian, kind of.
The main take-away here, though, is that you shouldn’t feel limited by what the books say. The books aren’t trying to restrict you; they’re trying to help inspire you, and give you a possible approach to a character. You’re allowed to change things up. So your “barbarian,” according to the books, is actually a “samurai,” in terms of how he thinks of himself, his social standing and training, and what he will respond to. If someone were to actually call him a “barbarian,” he might well fly into a rage at the insult—and some might see that as proving their point, while others would feel that a true barbarian wouldn’t find it offensive, and wouldn’t show the same appreciation for the finer things in life.
D&D 5e also includes backgrounds, separate from your class. You’re allowed to put together any combination of class and background you like. For example, you could use the noble background to reinforce that your barbarian isn’t actually barbarous at all. After all, Jack was a prince. Or you could use sage, since Jack was well-educated and much of his time was spent researching a way home. These background can help talk about your character beyond being a typical version of your class.
But you can really bring this to life by making appropriate class feature choices with your character, too. For example, at 3rd level, a barbarian must choose between the path of the berserker, and the path of the totem warrior. The berserker becomes almost mindless in a rage—not a lot like Jack. Meanwhile, the book says that “The Path of the Totem Warrior is a spiritual journey, as the barbarian accepts a spirit animal as guide, protector, and inspiration.” For a samurai, the “spirit animal” may actually be an ancestor—possibly an ancestor known as the Bear, the Eagle, or the Wolf in life, and that is why he or she returns in that form now.
The book doesn’t say anything about the spirit animal being an ancestor. That’s a detail I just made up. But making up details like that is how you turn a class in a book—a kind of proto-character the book’s authors made up—and turn it into your character.
So if you are looking for a simpler approach to your first game, and want to play a character like Jack, I recommend sitting down with your teacher and saying you want to play by the book’s rules for a barbarian—but a “barbarian” who is actually a samurai.