Is there a way of tying characteristics to a custom race rather than the character background? I was wanting to add specific personality traits common to the homebrew race I am working on. If not, is there anyway of making a homebrew background and tying it the the race like a link without DDB getting angry that there's a link in a description?

DDB doesn’t really care if you put a link in something, it just won’t let you publish it. But you can put in the link and use it privately all you want.

All you really need to do is create a trait to your race named “Characteristics,” check the “Hide In Sheet” box, and embed a table with the characteristics listed on it. The player can then simply copy/paste their chosen characteristic(s) into the appropriate field (Traits, Ideals, Bonds, Flaws). Those are form fillable, people can enter anything they want in those fields, they don’t have to choose from the ones listed under backgrounds. Heck, the newer backgrounds don’t even come with or-generated characteristics.