Heading north on the 101, take the Hollywood Blvd. exit. Take a right and an immediate left onto Van Ness, then take a left onto Canyon Drive. Take Canyon Drive past the sign that reads "No Access to Hollywood Sign", to the end where there is a small parking lot. From there you can access the Brush Canyon Trail.
There is another trailhead from Mulholland Highway. While it is closer to the sign, it is harder to get to by car.
My wife and I started from the gate at the Brush Canyon Trail. (Note that all the names used here are those that appear on the Google map of the area). We headed northeast on the wide dirt trail until it dead ended into the Mulholland Fire Road. The climb was moderately steep, but easy. We then headed east (left) on the windy Mulholland Fire Road. It was a nearly flat and was also a wide dirt trail. This part of the trail then dead ends into Mt. Lee drive which is paved. We then headed north (right) on this road which was quite steep until we reached the back side of Mt. Lee. From the north face of Mt. Lee there is a great view of the valley. As we made the final hairpin turn at the top of the Mt. Lee Trail, you could finally see the back of the Hollywood sign, but until you climb the final bit of trail to the top of the Peak, you cannot get a view of the sign unobstructed by the chain link fence. From the top of the peak you get a nearly 360 degree view of LA and the valley. The smog at noon was disappointing, so you may want to wait until later in the day to make the climb.
There have been many arguments about the presence of police helicopters at the Hollywood sign. On our trip we did see a police helicopter hover by the sign for a minute, and then leave. There are stories about cameras and infrared sensors, but the most logical explanation seems to be that the police helicopters just make the Hollywood sign part of their normal routine. While we there, they came by twice, and each trip was about an hour apart. There was no one down by the sign, so they just hovered for a minute and then left.
There are many Neighborhood Associations and websites that would have you believe that you cannot hike to the Hollywood sign. Well, you can. Even the official website of the sign says "Many, however, are surprised to learn that it's illegal to hike to (or basically get anywhere near) the sign." This is untrue as you can see from the attached photos. There are fences at the top of Mt. Lee to keep people from getting direct access to the sign, but even these are easily bypassed at the far east end. In fact, the only way to get to the top of the adjacent Cahuenga Peak (1820') is to bypass the fence. All of the trails leading to the sign have free public access.