I am coming back to DnD after 20 years of not playing. I have had a character in my mind and have had a back story that I have been thinking of for years now, and I can't figure out how to build the character. His father was an Elf Cleric and his Mother was a Dragonborn Warrior. His mother was badly injured in a battle and was cared for by his father, they fell in love, and as such my Character came into existence. However, there seems to be some prejudice in the world against an elf and a Dragonborn having a kid. As in it seems Half-elf has to be also Half-human. OR that is at the very least what I have found in the last 3 character creation tools that I have attempted. Any ideas? I'm not trying to break any game rules, I have put thought into the character and how the 2 extremes will play out in his behavior and gameplay. To take away his Dragon half would change his personality greatly in my mind. 

Tasha’s has rules for custom races. You might check those out for how to handle this one, mechanically, at least. That said, there are people who really don’t like tgose Tasha’s rules, so check with your DM, it’s a genetic answer, but it’s also the best. 

The rest you can probably manage though role play, never feeling quite like you fit in.  

It's not so much prejudice as it is an assumption elves and dragonborn are biologically incompatible. If you want to change that for your setting or the DM is ok with half-elf half-dragonborn characters, it's totally fine to do so. You can repurpose an existing race (including someone else's published homebrew), use the custom lineage option or homebrew something new yourself. 

Yeah, what Pang says, it's not a culture thing per se as a rules as written.  "Half Elf" as defined in the PHB is of elf and human parentage.  So without going into homebrew, you're best option is the custom lineage feature available with Tasha's (or maybe just available on this site post Tasha's).  If you want some sorta mash up of Elven traits with a breath weapon, that's going to have to be homebrew.  

There's some great third party stuff that talks about "unconventional" (that is not in the PHB or officially supported in any books) parentages but to translate those from those source's guidance to DDB would require homebrew.

Dragonborn are meant to be pretty rare. Elves aren't as rare, but certainly aren't as common as humans. As a result, the odds of having a Elven-Dragonborn hybrid is pretty low due to pure statistics. That's assuming that it's biologically possible - strictly speaking, it really shouldn't be, but it's fantasy so whatever.

As a DM I'm pretty reluctant to use Tasha's rules generally, but I'd be willing to use them for this. The only concern would be dragonbreath - it's the main thing for DBs, but too powerful to have plus things from the Elven heritage. Maybe the damage could be reduced or something to help balance it.

The Half-Elf statblock is already pretty versatile... you always get a +2 to CHA, but you can pick any other Ability Scores to increase that match your character concept. You also inherently get to pick two skills that could match your concept quickly and easily as well, and the class gives access to the languages of Common and Elvish, plus one more of your choosing, which could be Draconic. At that point it's mostly just flavoring on your part to describe your character, and depending on the class and subclass you choose you could use that to fully realize your character.

The obvious choice is to be a Draconic Bloodline Sorcerer... this gives you some of the direct benefits of being Draconic in nature, and overtly states that part of the Subclass manifests as scales on your body. Just learn the Dragon's Breath spell and bada-bing bada-boom, you got yourself a Half-Elf-Half-Dragon.

All that said, Homebrew is still always an option. I'd probably just recommend removing one of the features of being a Half-Elf and replace it with the Dragonborn feature you want most. Like... maybe drop the Half-Elf's Skill versatility and replace it with a breath attack.

Thank you all. Each of these has given me real information and direction. I will try out Tasha's Rules and look into a Homebrew option. I am still well into my character-building phase and This character is going to be my only planned build with several campaigns in mind. I have 2 different friends who do DM for different campaigns, one of them is excited about where my character is going, mainly because she likes the depth of the backstory and attention to detail that has gone into not only the race of the character but the complicated life that has created this characters Rouge, charlatan, chaotic good personality and skill set. 

Custom Lineage, as others have said, would be a good option. Or depending on what you are looking for out of the race, as far as traits, you could just pick half-elf, Dragonborn, or Elf, use those stats, traits, etc. and RP the half Elf, half Dragonborn aspects. Nothing keeps you from picking Half-Elf, for example, and say they have scaly elven features. Or picking Dragonborn, and describing them as draconic yet slim with pointy ears and a breath weapon. 

And home brew is always an option. 

Quote from Linklite >>

Dragonborn are meant to be pretty rare. Elves aren't as rare, but certainly aren't as common as humans. As a result, the odds of having a Elven-Dragonborn hybrid is pretty low due to pure statistics. That's assuming that it's biologically possible - strictly speaking, it really shouldn't be, but it's fantasy so whatever.

 

 Sorry, I just noticed I didn't explain why I wrote this - my point was that they don't include the mechanics because they're meant to be so rare, not necuase they're impossible or whatever.