When you do something in an informal or very personal way, you do it familiarly. Your grandma might greet you familiarly by saying, "Hi, sweetheart!"
Something that's familiar is either well known or very informal and intimate. The adverb familiarly describes things that are said and done in a familiar way. If the news reports on something in a familiarly sensationalist way, it's a type of reporting that you see all the time. But if your best friend hugs you familiarly, he does it lovingly. It's not surprising that familiarly shares a Latin root with family.