I looked around and couldn't find it in the articles. Is there a way for a character who is a cleric to add a second domain in the character builder? One of the players in my campaign is a cleric who has 2 domains.
You cannot by RAW, no. Multiclassing is built for multiple classes, so the rules (and thus DDB) have no support for building into the same class more than once.
My recommendation would be to make a homebrew version of one of the domains using the subclass creation tool, and add into the level progression of that domain any additional features that have been granted to the character.
Thank you very much for the feedback. I have been gone for a long time from D&D (I did play a little 3.5 but really AD&D is the last serious go around). I love 5th, but things are different enough it throws me off.