Luckily, my wife and I were there in the offseason (beginning of March) so there were not too many crowds to speak of. If you are willing to walk 15-20 minutes in either direction you will find yourself alone and be able to enjoy the scenery in peace.
After we got to an isolated spot we found a bench and spent an hour just taking it all in. That afternoon was definitely one that we will never forget.
I'm sure it had plenty to do with the section of the wall we explored; Simatai, a 5.5km section of the East Wall, was quiet by tourist standards, beautiful in its ancient way, and characterized by the immense grandeur of the rolling countryside. I thought the fact that we got to ride a zipline off the wall - and over a massive lake - was pretty neat.
This section of the wall was closed in 2010 for repairs, but could reopen anytime. If you'd like to speed things along, I'm sure my friend here would be happy to hire you.
The secret? Either head there in the off-season, or go in the late afternoon. In this case we got to the Mutianyu section on a weekday in May around 2pm. We stayed until the cable car down was about to close - roughly 5pm. We passed a few people, but most of the time we didn't see anyone else.
Other than the timing, the smartest thing that we did? We bought a Chinese robe and two umbrellas at the base of the wall for my cousin's wife. With no one on the wall, the photos wouldn't have been interesting otherwise.
Full account here: http://aliscottwhatwegetupto.blogspot.ca/2012/03/great-wall-of-china.html
Hiking the Great Wall was an amazing experience. What made it even better was going to Simatai, which isn't as heavily visited as Badaling and other parts of the Wall. It's harder to get to and takes more effort to reach, but it's worth it. The Wall stretches out for miles and miles. Totally cool.
Be sure to take the toboggan down this UNESCO World Heritage site, and stop at the School House on your way out of town for some delish food and an on-site glassblowing studio. http://www.theschoolhouseatmutianyu.com