I've had a couple of mentors in my career and I know people who actively engage with their own mentors.
First of all, successful people have a lot of people looking to them to be their mentors. So you're probably part of a long list of people. Your boss most likely doesn't have the time to respond positively to all these requests, so it's up to you to make it worth his while. The mentors I have had have all appreciated my efforts in that regard.
To engage successfully with a mentor, I'd first of all find a way to make his stress less (and not more). A pastor in my church had a guy volunteer to mow his lawn in exchange for mentoring. I don't think you need to go into providing free lawn care, but the principle is a good one: how can you make his life easier? That gets missed a lot. Part of that, especially in the workplace, is to be the best you can be at your job. If you stink at it, your boss won't be interested in helping you advance your career. If, on the other hand, you are excellent at your job and make his life easier, that makes the prospect of mentoring you more appealing.
Secondly, beware of the time commitment you ask of a potential mentor. You're looking for advice, not a career map. Weekly sessions with this person would be very draining to both of you and makes you seem incapable of working without your hand being held.
Determine where you want to go and what you think of the best path to get there. Mentors don't want to do your work for you; showing independence and the ability to do your own research will help a lot. The mentors I have had have all appreciated my efforts to improve THEIR performance, which led to their giving me more responsibilities and exposure to more interesting work. It was definitely a 2-way street.