NB: All information posted in my answer is based on a blog post by Alexander Lapshin. I have not personally visited Palestine.
The West Bank is divided into three zones:
Zone A (red on the map) is under complete Palestinian control. It is officially forbidden for Jewish Israeli citizens to visit Zone A without a special permission, out of concern for their own safety.
Zone B (light green) is under joint Israeli-Palestinian control. Israeli citizens may visit this area.
Zone C (dark green) is under full Israeli control. This is where the highly disputed "Israeli settlements" are located.
Entering Zone A
Practically speaking, border control is usually very lax. Israeli soldiers posted on the border of Zone A mostly care about explosives in your trunk rather than tourists sneaking in. You will see the following signs when entering Palestine:
The sign warns you that it is forbidden for Israeli citizens to proceed.
Legally speaking, the law states that Jewish Israeli citizens are forbidden from entering, while everyone else is allowed. The law is meant to allow Arab Israelis to visit their relatives, but this creates a lot of loopholes as there isn't a clear legal definition of who a "Jew" in regards to Zone A. For example, many Russian Israelis are Jewish by descent but not Jewish religiously.
Disguising your country of origin
It is best not to enter Zone A on an Israeli car or a Palestinian rental car. It would be obvious that you're Jewish and the soldiers would ask you to leave.
The most practical way is to park your car in Zone B and then catch a mini-bus to Zone A on the nearest highway.
Getting caught by soldiers
Since Zone A is patrolled exclusively by Palestinian militia, you may only get caught outside of Zone A. In this case one should never admit guilt and deny ever setting foot in Zone A, even if it's obvious you've just been there. In the worst case scenario demand to see a lawyer.
Public transport to Zone A
There are two options:
Take a public bus from Jerusalem to Abu Dis (in Zone B). From Abu Dis take a public bus to Zone A. This way you avoid going through Israeli border control.
Take a public bus to an Israeli settlement (such as Modi'in Illit). Then walk outside the settlement and catch a mini-bus going to Zone A. It is recommended to hike a few hundred meters away from the settlement before catching a bus.