You cannot use the object pronoun "me" as the subject of a sentence - you should use "I". I would also reverse the order of "Besides your father" and "I", as this sounds more natural to me. I would also change the tense to present perfect where you describe something happening over the past few days. And to answer your question, you should use think, not thinks. thinks only works for the third person singular, and you are talking about yourself - this is the first person singular. This is how I would phrase it:
Besides your father, I think that your attitude these past few days has been unbearable and very rude indeed.
However, if you switched it so that "your father" was the main subject, you would use thinks:
Besides me, your father thinks that your attitude these past few days has been unbearable and very rude indeed.
In the above case you would use "me" because it's the object of the preposition "besides". In general I might use along with instead of besides in either of these sentences, but that could just be my personal preference.