It isn't overpowered, but it is strong. For example, it allows a Monk to make 3 attacks per round at level 1, or 4 if they are willing to spend a ki point. It's also better damage than any other weapon, as it works out to be (5 + Str * 2) on average vs (7 + Str) for a greatsword.
It's also great for any character that wants to make as many attacks as possible. As an example, Paladins are already known for their nova rounds - now a level 2 Paladin can do (1d4 + 2d8 + Str) * 2 in a single round. It will burn up all their spell slots for the day, but against a powerful monster that they really want to see dead it's definitely going to accelerate their victory.
But where it becomes really problematic is at level 5, when classes start getting Extra Attack. With no interaction specified, this will allow 3 attacks per round for any martial Dragonborn, or 4 if they have a way of getting bonus action attacks (5 for Monks who spend a ki point). Thankfully, this doesn't work with some of the best methods of adding extra damage to every attack, but it will still enable some incredibly high-damage builds.
At level 10, things become problematic all over again, as now every Monk wants to be a Dragonborn, and the claws deal better damage than any one-handed weapon while also enabling more attacks.
To make this non-broken, you need to at least make it require the use of a bonus action, and fix up the interaction with Extra Attack. However, you've said that this feature is in addition to a Dragonborn's normal features. Therefore, rather than making it still strong but not broken, you might want to aim a little lower. I would recommend modelling it off a feature that already exists and represents exactly what you're trying to do:
Talons. You are proficient with your unarmed strikes,
which deal 1d4 slashing damage on a hit.