There are three main parts to your character, their background, their race, and their class. Another way to look at this is, where they came from, what they are, and what they do. Once you have these three things picked, you write a backstory that ties them all together and there’s your character.
One problem I have with 5E, is when these three parts are needlessly overlapped. For an example of what I mean by this, lets look at the Monk class. The class description essentially starts off by saying that your background is that you lived in a monastery. Now, I understand that a lot of people hear monk and think of either Shaolin priests or British guys in robes with bad haircuts, but what exactly is a monk?
So, what exactly is a monk then? Essentially, they’re someone who trains their body for hand to hand combat to such extremes as to gain what appear to be magical abilities. You don’t need to live in a monastery to do that. Looking at the real world, we have places that will teach martial arts in malls all over the country. You can take lessons at gyms. You can even get a DVD and train at home. This means that your monk doesn’t necessarily have to be from a monastery. They could have trained to fight in dockside matches at Waterdeep, or they could be a thief who trained to fight because getting caught with a weapon had much stiffer penalties than just getting caught stealing something.
An example of a non monastic monk would be the character that inspired the class, Remo Williams, from The Destroyer series of books, the movie Remo Williams: The Adventure Begins, and the tv series that it spawned. The titular Remo Williams is a cop framed for murder whose death is faked so he can become a secret agent and assassin, all while living in his master’s basement apartment.
The same goes for the barbarian class. The flavor of a barbarian coming from a distant land or living among a primitive tribe isn’t needed for the class. A barbarian is essentially a warrior that uses rage to fuel their attacks. A great example of a barbarian from a civilized land would be Ser Gregor Clegane. Every time he fights, something pisses him off, and he gets stronger. A noble totem barbarian could even be a knight calling on what ever animals make up his family’s coat of arms for power.
So, when making your character, remember, your class is just a description of how your character gains their power, not where they came from or what they did before becoming a character. A knight doesn’t have to just be a paladin or fighter. They could be anything from a ranger to a warlock. It’s your character, you can do what ever you want, so go ahead and have fun with your class and background.