Both phrases have different usages in different contexts, but when expressing persistence of an action or state, "all the way" implies some movement, such as in the sentence "He was chewing gum all the way to the store." "Throughout" emphasizes the encompassing scenario or situation, such as in the sentence "They were talking throughout the whole movie."

I have heard of doing something all the way ... being something all the way, though I think it's slang. But I haven't heard it expressed with "throughout." Unless you mean someone "did it throughout his life" or "throughout the entire movie" etc. Maybe "through and through." To me, through and through is a little old fashioned. These expressions are found in the idiom reference sites. She's a Yankees fan all the way. A fanatic. She's a Yankees fan through and through. Really committed. When he does something, he does it all the way.