I have a small family/my family is a small family of three. I, my wife and her brother (or a brother-in-law instead of her brother -have the same effect, though.)
My name is Renan and my birthday is on July twenty-third (or written “on July, 23.”).
My wife´s name is Maria, and her birthday is on October twenty-fourth (ditto here).
My brother-in-law´s name is Fabiano, his birthday is on August twenty-sixth.
Let´s wait more comments.
Overall, your piece of text was good.