The title really explains itself mostly.

So, what I mean here really is how, in dialogue and the characters themselves, just how to make them not human. For example, the struggle with elves in fantasy is that most often their just humans with pointed ears: they don't act differently, talk differently, feel different, etc. Most often, when people do things with elves they just feel like humans to me, because there's nothing really different from them.

I'm trying to create sort of my own dragonborn race (obviously a less nerfed one) and I'm trying to make them more unique. For example, the main character is dragonborn that the players need to make king, named Uthur II (how I can make him an interesting character is something I'll need help with, but that's for another day lol), and I want him to stick out.

honestly idk but it's up to you, they can  - have nontypical cuisine which human can't eat, different sleep cycles, different pronounce or idk talking in lowest form of basso, dragonborn can have different cultural values, and respect some qualities more or less than average human of your setting do, the way and the sound of their breathing can be different, smell differently, feel odors differently, see colors differently (in a way humans will never see or even understand), they can blink way rarely/oftenly or eg with very slow speed, have weird rituals and have sound perception differently, have non typical walking pattern, have totally different law system, have niche diseases/immunities, substance tolerance, be at different technological level and so on:

also from the start they already look quite differently so you have one checkmark from the start

even 3 well implemented physical difference checkmarks can be enough for a writer to win Hugo award like my favourite author Zelazny did with "this immortal", so it's basically up to your delivery of those differences

First of all, you probably can't to any great degree. You're a person, talking to people, and we're great at anthropomorphizing everything. Writing convincing aliens is extremely hard, and you're making up the dialogue on the fly, so you have no time to revise it to get the effect you want.

I've done something similar in a past campaign (played a Great Old One Warlock who was thoroughly losing their grasp on the concerns of mortal beings, and my fellow players agree I managed to make the character creepy AF), and I have some thoughts:

• You need several points of difference. Just one is more of a quirk than alienness, and will also get repetitive as you reinforce it.
• Some of them should be about how they physically interact with the world, and some should be personality 
• Figure out how to work the physical ones into describing their actions, and do so regularly. 
• This is especially important with things they don't do. If they don't blink, how do you occasionally remind your players of that?
• For personality stuff, figure out if it's biological or cultural. (Think Star Trek Vulcans. They are emotionless, but that is a learned behavior. It slips a lot. If it were biological, it wouldn't.) Also, it's at least partly cultural, no matter what it is.
• Some of the differences should affect their speech in a way that's easy to do consistently.
• Speech changes like hissing or growling is probably less effective in making them feel alien than word choice, grammar, or affect. (Never use personal pronouns, or perhaps pronouns at all. Don't use contractions. Be singularly unconcerned with the possibility of death. Etc.)
• The same basic trait should manifest differently, and to different degrees, in different individuals. Elves ought to have a very different attitude to time than humans, but in one, that might be "everything but the most pressing crises can be put off, thought deeply about, and discussed. We have all the time in the world. What does it matter if we set out now, or in a year or two?", while another might be "Anything I'm doing with these humans must be done now! now! now! If we don't go to get pizza from that new place today, it might close! Or the humans could die!"
• For cultural traits, if they encounter a large number of these people, remember that they're going to have more than one culture. Also, not all traits will be consistent within a single culture. A people with a strong honorable warrior tradition is going to have warriors who mostly pay it lip service. Also, they still need to have bartenders, tailors, and sketchy merchants who hang out in dingy bars by the docks, none of whom are going to react to a slight the same way as Sir Honorable of Honorare,

That's fair. I'm trying to write the entire story around these as well, so I thought more in-depth stuff would be interesting

Quote from AnoruxTheDragon >>

So, what I mean here really is how, in dialogue and the characters themselves, just how to make them not human.

 I think, in general, this doesn't happen. Like, at all. 

Take the dwarves: They're not human. But the best we've come up with is they speak in a scottich accent, and behave like slightly more belligerent humans. 

And the problem is that they're not really different from humans. Maybe they lay eggs? But that doesn't impact gameplay. So basically, they're scaly humans.

Think instead of a race that's actually different. Formians, for instance. They are a hivemind. Much easier to play that as actually not human: All formians in a group react as one, maybe speak as one. They have no individuality (at least the lower castes), and will gladly work themselves to death is the need presents itself. Maybe even unless specifically made to not do so. 

Or the mind flayers. Not a hive mind - but always in contact with the elder brain. And all that ancient, deep sea intelligence, powerful and fascinated with the workings of lesser lifeforms, entirely bereft of any moral concerns what so ever. 

Beings that on the basic level aren't human, are easy to play as such. Dragonborn - by design - are essentially scaly humans. Also I hate them. But if I had to use them, I'd settle on an asian inspired culture - samurai inspired warrior culture, caste divisions based on color and age, and so on. They'd be not so much unhuman, as simply foreign.