My boyfriend is sick and I'm frustrated with his constant bad mood.
Is there a reason he's in a bad mood all the time?
He's always sick and that makes him upset.
It is understandable to be frustrated by your boyfriend's bad mood.  However, it is difficult to be chronically ill.
Sometimes his mood makes me wonder if he even loves me.
Your frustration is understandable, but you should try to be more mindful of what your boyfriend is experiencing and try to see it from his point of view. I'm sure he loves you and perhaps the two of you should have a conversation about how his illness is impacting the two of you, both as a couple and individually.