It is possible to be an half dragon, but it requires a considerable amount of work and cooperation with your DM.
Advice: From what you wrote in this question, in the comments, in chat and in another question, it seems to me that you are a newbie in 5th edition of DnD: I kindly suggest to read the Player's handbook, in order to have a general sight about the rules. Moreover, I suggest then to bring this character idea to your DM and work it with them, in order to achieve a playable and balanced character.
Your DM may adapt the Half Dragon template in the Monster Manual: from the description of such creature we can read:
When a dragon in polymorphed form mates with another creature, the union sometimes produces half-dragon offspring. A creature might also transform into a half-dragon as a result of a mad wizard’s spell or a ritual bath in dragon’s blood. In all these cases, the result is a creature that combines the essence of a dragon with the form of its original race. Regardless of their origins, all half-dragons have similar features and appearance, gaining special senses, resistance to destructive energy, and a breath weapon.
It gains more senses, resistance and the breath weapon related to the lineage of the dragon bloodline (chromatic or metallic). Using this template, the DM may add more details and/or features, as for example the usage of wings once a certain level is reached1, or looking more dragon-ish a limited number of times per day.
Nonetheless, the DM has to take into account that this template covers the creation of monsters/NPCs, that usually follow different rules with respect to player characters's one.
Becoming a dragon X times per day will be quite unbalanced.
I would pay particular care in allowing becoming a true dragon: in the first tiers of play this could be really unbalanced, dragons are powerful creatures2, as Illustro notes in their answer.
There are some spells (True Polymorph, Draconic Transformation) that allow to become a full dragon, or even magic items (e.g. Potion of Dragon's Majesty, as stated in this answer to the other question of yours), but they are powerful and available at high levels, they are not suitable to low level characters, particularly for 1st level ones.
1 Check for comparison Drows or Tieflings, that gain the usage of a 2nd level spell once the 5th level is reached, regardless the class.
2 Well, we are playing Dungeons and Dragons...