This is kind of a two-part question: 1) If you say "你如何喜歡你的菜?" are you asking, "how does the food taste to you" or "how do you prefer your food to be cooked"? To me it seems that if you are asking the first question it would be better to say "你的菜怎麼樣" and if you are asking the second question you should say "你如何喜歡你的菜". Or is it basically the same question either way? 2) Is there a difference between 如何 and 怎麼樣 in either meaning or usage? I feel like there is but I can't quite define what it is. (If the second part of this question is to broad, I'm happy just getting an answer to the first part.)
你如何喜歡你的菜was machine translated for "How do you like you dish?" where "how" can be translated to either 如何 or 怎么样. As Drunken Master pointed out, 如何 is a bit formal and are more likely to be used in written, while 怎么样 are usually used in speaking.
My first thought of 你如何喜歡你的菜 was "How do you like your dish to be cooked?" as I just ordered my steak. And, medium rare please...:)
I feel that what you are asking is pretty much the difference between 怎么 (colloquial for 如何) and 怎么样. The difference is that 怎么 is used in a verbal predicate sentence, and 怎么样 for an adjectival predicate sentence:
你怎么喜欢你的菜？How do you like your dish?
你的菜怎么样？ How is your dish?
Just change 怎么 to 如何 and you have the same result in a more formal way.
"你如何喜歡你的菜?" more sounds like "What makes you like your food?". We rarely say that. I prefer Pedroski's answer.
如何 and 怎麼樣 are nearly the same. When you ask 2 "how"s, you can use 怎么样 for the first question, 如何 for the second one. Such as: "How about this cell phone? And how about its quality?" = "这个手机怎么样？质量如何？"。