I'll answer your question in two separate posts.
In a needs analysis, you ask questions to find out what students want to study in class. In your case, it will be very easy to deal with the results, because you have only one student. It's far more difficult when you have a group of people who all want different things.
You can do a needs analysis via a written questionnaire or through informal questions in class. I think it's best to use informal questions with your quiet teenager. Ask about his reasons for learning: how he will use English in the future? how does he use English now? Also ask what he wants to spend time on in class: speaking, listening, reading, writing, grammar, vocabulary, pronunciation etc.
When planning subsequent lessons, bear in mind what your student wants. Remember to get a balance in your lessons. Your student might want a lot of fun activities but it is up to you as the teacher to also include activities that will help him progress.