Familiarize yourself with as much of the topic you are studying that you can. Ideally, before approaching the expert, you need to have clear, defined questions. The less vague and ambiguous you are, the better and more informative your conversation with the expert will go, as you will have more credibility and not waste his time.
Go to the library or look online for any community lectures open to the public. For example, a small college in your area may be hosting a guest speaker who will be discussing the area of history you are studying. Most of these forums have open question and answer sessions that can prove useful.
Ask your instructor if she has any contacts in the field that may be of additional help to you. Don't worry about her taking offense, chances are she will be pleased that want to enhance your understanding of the subject by seeking out supplemental knowledge.
Write down several questions to ask. The more in depth your query is, the more you will learn. Remember, you are seeking out an expert because you wish to learn more than the standard information in your textbooks or at online expert sites. Have as many questions as possible on hand in order to best optimize your time with the expert. Be mindful of new questions presented in his answers. For example, if he informs you that King George III was not England's longest tenured monarch, ask who was.