Beyond simply wishing, polymorphing at a high level, there are some baby steps...

You're the DM, so anything is possible. But if you're looking for something that fully fits in the game's mechanics, this Unearthed Arcana specifically mentions three new feats that let a Dragonborn be, well, more draconic

If this is the sort of villain that 'grows with the party' these could make stepping stones to potentially wishing/polymorphing to be a full dragon

###Dragon Fear Prerequisite: Dragonborn
When angered, you radiate menace. You gain the following benefits:

• Increase your Strength or Charisma score by 1, up to a maximum of 20.
• Instead of exhaling destructive energy, you can roar and expend a use of your breath weapon to force each creature of your choice within 30 feet of you to make a Wisdom saving throw (DC 8 + your proficiency bonus + your Charisma modifier). A target automatically succeeds if it can’t hear or see you. On a failed save, a target becomes frightened for 1 minute. If the frightened target takes any damage, it can repeat the saving throw, ending the effect on itself on a success.

###Dragon Hide Prerequisite: Dragonborn
You inherited the might and majesty of your dragon ancestors. You gain the following benefits:

• Increase your Strength or Charisma score by 1, up to a maximum of 20.
• You grow retractable claws from the tips of your fingers. Extending or retracting the claws requires no action. The claws are natural weapons, which you can use to make unarmed strikes. If you hit with them, you deal slashing damage equal to 1d4 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
• Your scales harden; you gain a +1 bonus to AC while you aren’t wearing armor.

###Dragon Wings Prerequisite: Dragonborn
You sprout draconic wings. With your wings, you have a flying speed of 20 feet if you aren’t wearing heavy armor and aren’t exceeding your carrying capacity.

All of these give a plausible path to your villain becoming increasingly more draconic as they gain in power.

Beyond simply wishing, polymorphing at a high level, there are some baby steps...

You're the DM, so anything is possible. But if you're looking for something that fully fits in the game's mechanics, this Unearthed Arcana specifically mentions three new feats that let a Dragonborn be, well, more draconic

If this is the sort of villain that 'grows with the party' these could make stepping stones to potentially wishing/polymorphing to be a full dragon

Dragon Fear

Prerequisite: Dragonborn
When angered, you radiate menace. You gain the following benefits:

• Increase your Strength or Charisma score by 1, up to a maximum of 20.
• Instead of exhaling destructive energy, you can roar and expend a use of your breath weapon to force each creature of your choice within 30 feet of you to make a Wisdom saving throw (DC 8 + your proficiency bonus + your Charisma modifier). A target automatically succeeds if it can’t hear or see you. On a failed save, a target becomes frightened for 1 minute. If the frightened target takes any damage, it can repeat the saving throw, ending the effect on itself on a success.

Dragon Hide

Prerequisite: Dragonborn
You inherited the might and majesty of your dragon ancestors. You gain the following benefits:

• Increase your Strength or Charisma score by 1, up to a maximum of 20.
• You grow retractable claws from the tips of your fingers. Extending or retracting the claws requires no action. The claws are natural weapons, which you can use to make unarmed strikes. If you hit with them, you deal slashing damage equal to 1d4 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
• Your scales harden; you gain a +1 bonus to AC while you aren’t wearing armor.

Dragon Wings

Prerequisite: Dragonborn
You sprout draconic wings. With your wings, you have a flying speed of 20 feet if you aren’t wearing heavy armor and aren’t exceeding your carrying capacity.

All of these give a plausible path to your villain becoming increasingly more draconic as they gain in power.