I'm interested in visiting the Great Wall of China, but don't want to face many tourists during my trip.
Are there such parts of it?
Are visitors allowed to walk on it?
As with everything else in China - if you go to a place that is opened for tourists, be prepared to see a lot of them. You can usually avoid the hordes by going on a weekday (as opposed to weekend), avoiding national holidays, going early in the morning and picking a place that is less-known (pick #3, #4, ... place as opposed to #1, #2 place). That being said, the great wall is a very popular destination and the virgin places are harder to find.
But - being that long (6500km?) - there is logically quite a lot of choice if you're looking for a more unique experience than the "Disneyland" you see at Badaling or Mutianyu.
Here are a few options:
As for the second question:
You are allowed to walk on the sections opened to the tourists. Besides that - walking or camping on the wall is technically illegal, but you will probably have no problem (worst case you get a fine). And hey, camping on the great wall is awesome!
There are a great many locations on the Great Wall that can be visited. Narrowing the choice to those within day trip or single overnight stay from Beijing, we get the following list. I've added my comments as to their merits.
Simatai ( 司马台; Sīmǎtái) a popular but remote. Quite far from Beijing. Currently (2011) this section is closed to tourists but should reopen shortly.
Gubeikou is good but a little least visited, sections of the wall. Less tourists because it is more remote and further from Beijing than other sections.
Jinshanling ( 金山岭; Jīnshānlǐng), you can walk from here to Simatai. Arrange to have your driver pick you up at the other end. You can't continue on to Simatai as that section is currently closed.
Mutianyu (慕田峪; Mùtiányù) popular and accessable section of the wall. Favourite of photographers.
Jiankou (Chinese: 箭口，箭扣; pinyin: jian kou) highly photogenic and atmospheric section of the wall.
Huanghuacheng - very remote and little visited but quite attractive. A very steep climb.
Badaling (八达岭; Bādálǐng) This is a tourist trap. If you don't like crowds, don't go here.
Shuiguan - near to Badaling, again very touristy and busy in peak season.
Juyongguan or Juyong Pass (居庸关; Jūyōng Guān) easy to get to though heavily restored.
If you are willing to go a little further to really get away from the tourists and get a unique view of the wall, I can recommend Shanhaiguan (Shanhai Pass). This is located in Hebei Province about 300km east of Beijing. It will require an overnight trip to visit but you will get to see many interesting details of the wall without disturbance. This is traditionally the Eastern end of the wall, where it meats the sea. There is a large fortress here in addition to the Wall. A little further along the coast you can find the Dragon's Head where the Great Wall actually juts out into the sea. A unique sight.
The above information was sourced and edited from here.
You may find tours in Beijing offering to take you to "Secret" sections of the Great Wall. These may take you to areas that are not officially open to tourists. I would request that you avoid these tours. The Great Wall is fragile and easily damaged. The sections that are open to tourists are managed so as to prevent and repair damage. When visiting closed sections of the wall you are destroying the very thing you came to see.
Author Peter Hessler visited some of the most remote parts of the Great Wall and seemed to do an excellent job of avoiding other tourists. He took measures to avoid catching the attention of the local government, though. In planning your trip, you might do well to read his account of the journey: Country Driving: A Journey Through China from Farm to Factory