I assume you put question marks behind these sentences because you are asking us a question. Clearly, question marks don't go there, those two examples are not questions. "I look like my bother" is correct. You would use "looks" ONLY in the third person singular preset tense, with pronouns he, she and it. This is the only one where you need to add an "s" to the end of the verb.
I look / he looks
I go / she goes
We see / he sees
They fly / it flies
You write / she writes
When you ask a question in the present tense, you need to use "do" or "does" in the same manner. If it's a third person singular, use "does" and do NOT double up on the "s".
Does he look like his brother? Yes, he looks like his brother.
Does it fly high? Yes, it flies very high.
Does she write a lot? She writes every day.
Does he see that? I think he does, he sees that.
Do you look like your brother? I do, I look like my brother.