I am new to character creation and interested in creating a half-elf half-dragonborn character (e.g. pointed ears and elf-like face with dragon-like horns).
Would I pick an elf or half-elf race and just mention dragonborn in the backstory, or is it actually possible to have a dual race character?
There aren’t rules for doing that: the rules offer “half-elves” (and “half-orcs”) as a race, but in both cases they aren’t mix-‘n’-match options, but static races that always offer (more or less) the same features—and whose other half, in both cases, is always human. In other words, Wizards of the Coast figured out what someone with an elf and a human parent (or an orc and a human parent) is like, but they didn’t offer any rules for figuring out what someone with an elf and a dragonborn parent would look like.
So anything you do on this front is going to require the assistance of your DM, and they may just say “no,” either because they don’t feel confident making up rules for it, or because their setting, like the base setting, doesn’t allow these races to produce offspring together.
For simplicity, my suggestion here is to just say that one parent or the other “dominates” the character’s fenótipo, and you use the stats for that race—and the effects of the other parent’s race are purely aesthetic. So you make a dragonborn character with longer, more pointed ears, perhaps, and say they are half-elven. Or you make a half-elf character with unusual hair or eye color, maybe some scales, and say they are half-dragonborn. Even this will require DM approval, really, but I imagine most DMs will be more likely to approve this than anything more involved, simply because it’s easy and “safe.”
I was reading your question and had a crazy thought that could actually work out real well as far as balancing such a race.
In the Sword Coast Adventurer's Guide, the Tiefling variant race (SCAG, p.118) allows you to generally resemble a human (or maybe an elf?) with added horns and patches of leathery/scaly skin reminiscent of a dragonborn's. I've taken the liberty of quoting the relevant parts that allow this race to work.
TIEFLING VARIANTS
Since not all tieflings are of the blood of Asmodeus, some have traits that differ from those in the Player's Handbook. The Dungeon Master may permit the following variants for your tiefling character, although Devil's Tongue, Hellfire, and Winged are mutually exclusive.
Appearance. Your tiefling might not look like other tieflings. Rather than having the physical characteristics described in the Player's Handbook, choose ld4+l of the following features: small horns; fangs or sharp teeth; a forked tongue; catlike eyes; six fingers on each hand; goatlike legs ; cloven hoofs; a forked tail; leathery or scaly skin; red or dark blue skin; cast no shadow or reflection; exude a smell of brimstone.
Feral. Your Intelligence score increases by l , and your Dexterity score increases by 2. This trait replaces the Ability Score Increase trait.
Hellfire. Once you reach 3rd level , you can cast the burning hands spell once per day as a 2nd-level spell. This trait replaces the hellish rebuke spell of the Infernal Legacy trait.
Technically speaking, the variant tiefling race would allow you to look like a half-dragonborn. In the PHB, the examples of horns that are described vary tremendously. Perhaps yours look like a dragonborn's. If your DM also allows pointy ears for fluff (with tieflings, who knows?), you could definitely pass for what most people would expect a half-elf/half-dragonborn to look like.
Through the Infernal Legacy racial trait, you gain access to the thaumaturgy cantrip. Coincidentally, High Elves would also normally gain a cantrip! Thaumaturgy isn't normally available to elves, but you're a half-dragonborn, so it works out. If your DM is generous, perhaps he would consider allowing you to take Prestidigitation instead. Both are mostly flavorful in nature rather than being combat oriented, and some of their respective effects are rather similar, so it could work out fine balance-wise.
Elves normally get +2 to dexterity and +1 to intelligence. Coincidentally, the "Feral" Tiefling variant grants you the same bonuses. However, this means that other variant options are out of reach (other than variant appearance, they are all mutually exclusive), which is a bummer if you also want a "dragon breath" attack. On the other hand, regular tieflings grant you a +2 to charisma and a +1 to intelligence, which could pass of as the bonuses granted by the Half-Elf race (+2 charisma and +1 to two other abilities).
As a dragonborn, you normally gain resistance to your chosen element. Assuming you are OK with being a red, brass or gold half-dragonborn, who all share the fire element, the tiefling race also grants you fire resistance.
Furthermore, the Hellfire variant allows you to cast the Burning Hands spell once per day, which you could refluff as a dragon's breath attack. The damage is mostly similar, but yours would only reset after a long rest. But that's ok, you're not a "full" dragonborn after all! Since burning hand is a cone rather than a line, it would mostly fit a red or gold half-dragonborn, as brass dragonborns normally get a linear area of effect.
Dragonborns normally receive a +2 bonus to strength and a +1 bonus to charisma. You don't get a bonus to your strength, but that's ok, you could take after your "elf" parent more, as your chosen appearance would suggest. As for charisma, you get double the bonus by being a tiefling, which can be really nice depending on the class you want to play. As you mentioned considering Sorcerer as a class for your character, this works out real fine.
If your DM is OK with this, you could be a tiefling only on paper for game balance, but in reality actually be a half-elf/half-dragonborn.
Alternatively, you could give yourself a classic "unknown origin" background such as having been left as a baby on the steps of a church. You're actually a tiefling in this case, but you've been raised by adoptive parents who believed you to be a half-elf/half-dragonborn. That's what you look like, afterall!
Although, in terms of mechanics, it would be a new race, rather than a pair of two existing ones.
If I understand you correctly, you are interested in creating an engaging backstory in the first place, and 5e actually encourages such creativity.
Your game mundo does not have to be restricted by the PHB content. If your world have half-elf half-dragonborn people, and your DM agrees with that, the game permite you co collaborate and customize the character's race:
The most important step in customizing or designing races for your campaign is to start with the story behind the race or subrace you wish to create.
... begin with the story and proceed from there. Compare your creation to the other races of your world, and borrow freely from the traits of other races
See Chapter 9: Dungeon Master’s Workshop of the DMG for more details.
There is no hard rule that a "half-something" has to have the 'something' bit be human (it is usually assumed and implied, but not required).
Rule-wise though, you have to choose a single race, and then describe the other half in your backstory, giving you no benefit.
For OP's case; you can choose to be a dragonborn, and mention elf in the backstory in order to get the breath weapon and energy resistance; or you can choose to be half-elf, getting darkvision and the weapons, but you won't get both benefits.
"Ask your GM" always applies. At my table, I would ask you to think about what elements of dragonborn and elves are you looking for:
Homebrew creature/race creation is an idea that has been explored a fair bit among worldbuilding GMs and designers over the years. It comes down to figuring out (A) what you want, and (B) what you can build/repurpose from existing material.
You can find some guidance by checking out online discussions of 3e/4e Bloodline feats, Pathfinder Heritage feats, and the like, or check out various Dragon articles and third-party publications exploring building custom content.
Tags dnd-5e dragonborn elfo corridas