Using "抱怨": I'm really skinny, but I'm not complaining Not sure how to use "抱怨."
Would it be correct to say: 我很瘦，但我不在抱怨。
Or if there is a better phrase to use in this case, please let me know!
it is appropriate to use "抱怨" here. Your translation is also correct. But I think a better translation is "我真的很瘦，但我不是在抱怨" which more accurately reflect the tone of the English sentence. 

(BTW that was from Google Translate)