For the homebrew character of dragonborn goliath it's allowed the breath weapon choice but having trouble trying to add those choices on the character form. What are the ways to add it for building it?

If you are making a homebrew race, I would advise creating a copy of each and see exactly how it is put into the system and just choose one as a base to add whatever seems easier to that race. So, starting with dragonborn and adding the goliath stuff that you want to it.

Well when you go to create a normal Dragonborn character in the creation you choose what ancestry for the breath weapon. But when I go to the character creation how do I add those said choices to choose from that's pretty much what I'm asking.

If you are playing a dragonborn it is a dropdown that you choose your ancestry on. Dragonborn is a race, so you cant be both a dragonborn and a Golith unless you are homebrewing something.

If you meant draconic bloodline sorcerer, it functions the same. There is a dropdown that you can choose what type of dragon you came from. It will have a little blue ! next to it if you haven't made a choice yet. 

In either case, all of the dragons you have to choose from will be available to you. It functions the same way as choosing any other thing in character creation.

How do I add the said dropdown bar for the choices because I am homebrewing the dragonborn goliath and I've done research and it is a know race but I have to homebrew it myself so I'm trying to add the drop down bar for the creation.

This is where I said, create a copy of the race and see how it was done then duplicate whatever traits you want from the other race into the new one. When you create a copy of a race you can look at exactly how it was put into the system. 

I would also recommend looking into the Homebrew Section of the Forums where they have a lot of different guides for almost anything you may need. 

I've linked a good forum thread for you here.

Well thank you that helped me quite a bit