On this trailmap, you'd start from the Observatory, walking north, to the Charlie Turner trailhead, take the trail over Vermont, then take the thin, steep trail on your right (not the rutted use trail along the pipe) to meet up with the Mount Hollywood trail, south of Mount Hollywood. There is one trail junction along this route, where heading straight would take you down to Vermont and the "Bird Sanctuary, while left takes you up towards Mount Hollywood.
Once you rejoin the big dirt fire road, turn right and walk along the eastern side of Mount Hollywood, then continue to the "North Trail," towards Taco Peak.
[If you go right, along the west side of Mount Hollywood, it's my suspicion that it's slightly longer and with some wasted climbing, but the difference is not great.]
As you near Taco Peak, take the narrow trail along the big pipe up and around the north side of the peak. Once to the north of the peak, it's a wide dirt road that'll bend back around and take you towards Mount Bell. I typically follow the dirt road down and around the south end of Mount Bell, following that dirt road until it reaches the paved (but closed to public vehicles) Mount Hollywood Drive, then picking up as it climbs a bit up to the pass between Mount Chapel and Taco Peak. There, a narrow trail heads long the "ridge," towards the summit of Mount Chapel. Veer left before the steepest ascent begins and stay to the south of the peak. Follow that thin trail, with many ups and downs, as it runs south of Mount Chapel's summit. There may be quite a lot of brush, with lots of bees visiting thef lowers of that brush. Long sleeves and long pants could be helpful.
Just before you reach paved Mount Lee Drive, a clear but thin trail drops down and to the right, while the possibly clearer use trail continues along that last bit of ridge, before it reaches a dead end. If you come to the deadend, backtrack the few yards to the spur that dropped to the northwest. It hits Mount Lee Drive within 50 yards or so. Once on Mount Lee Road, walk the pavement up (right, intially) to the top of Mount Lee. You'll pass to the north of Mount Lee first, with Forest Lawn and Burbank to your right, along a long incline. When you make the hairpin turn, you'll see the back of the sign.
Had you continued on the dirt path straight ahead instead of turning with the road, you'd go to Cahuenga Peak, then on to Burbank Peak.
Continue past the sign a few yards, and trail takes you to an overlook, on your left. You can see and photograph the entire sign's back more easily from up there.
It's now late spring. Still some flowers in bloom, though not too many really thick blooms. Rattlesnakes are also active, though I didn't see any. The narrow "Ridge Route" offers some rattlesnake potential, btw. Sticking to the wider dirt roads would be safer from a rattlesnake perspective, though somewhat longer.
In terms of what flowers you're looking at in this post, in order, it's bush sunflower, bush sunflower, bull thistle, Indian pink, farewell to spring, California buckwheat, blue dicks, wild mustard, sticky monkey flower, golden yarrow, chaparral yucca, pearly everlasting, farewell to spring, phacelia, and morning glory.
As noted at the top of this post, I made the roundtrip in under three hours (about 2:45). This was with a slight wrong turn around Taco Peak on my return, which added about 8 minutes, I would guess. I've done this hike often, but not that recently, so I wasn't always sure I was on the right path, but that's the only wrong turn I made. Also, I'm a pretty fast walker by nature.
My Alltrails showed the total distance I covered as 6.12 miles, a gross altitude gain of 1158, and 2:11 of moving time (meaning, ignoring my stops for pictures or to sip some water), which is roughly a three mile an hour pace, which sounds about right. So, although it was less than three hours from the Observatory to the Hollywood Sign and back for me, if I wanted to stick to the wider dirt roads and rely on line of sight to choose my path because I was less sure of the proper route, I would definitely allow four hours roundtrip.
Just adding the Mount Hollywood Trail part in (rather than cutting up the steep designated trail that starts right after the Vermont Tunnel) adds at least two miles, roundtrip. Dropping down to the Mulholland Trail instead of the "ridge route" probably adds a similar distance. So then you'd be talking about 9-10 miles rather than 6 miles, which would be a non-trivial increase in length. The longer way also adds more gross altitude to be gained and loss.
Of course, if you just want to get to the Hollywood Sign, there are shorter trails to get there. But if you park anywhere directly south of the area between Burbank Peak and Mount Lee, be careful. The roads are narrow and winding, and the closest paved roads are highly restrictive in terms of parking.
This way works for me because I have convenient parking up near the Observatory. If I parked down at the Greek, that would add another mile or so roundtrip. Alternatively, if I took the free DASH bus up to the Observatory, the distance would be the same as parking up there.
If you're taking public transit to the Observatory, the Observatory LA DASH bus is free (as are all of the local circulator DASH buses), and runs between the Sunset and Vermont subway station and the Observatory, with several stops in between. This subway route used to be called the Red Line, but now is called the B-line, because it was the second transit line built, after the "Blue" or A-line, that runs between 7th and Wilshire ("Metro Center") and Long Beach.
If driving a car, from DTLA, you'd either take the Hollywood Freeway (US-101) north and exit at Vermont, heading straight north from there up to the Observatory, or you'd take the Golden State Freeway (I-5) north and exit and Los Feliz. Take Los Feliz west, and turn right (north) at either Hillhurst or Vermont. The two streets merge a bit up the hill, and continue on to the Observatory.
From the north, if on I-5, you'd still get off at Los Feliz and head west, same as above.
From the 101, you'd probably exit at Hollywood or Sunset, head east to Western, then north, until you merge with Los Feliz. Then Los Feliz east, to Vermont.
If you just want to get to the front of the Hollywood Sign or to Burbank Peak, you'd be able to get a lot closer by using one of the trailheads closer to Lake Hollywood Park. To get there, I'd exit the Hollywood Freeway from Cahuenga, and take that to Barnham, then head "east," then turn "east," again, on Lake Hollywood Drive. [The streets don't run in the cardinal directions around here. There's a traffic light at Lake Hollywood Drive that just says "Lake Hollywood," and there's only one way you can turn, there.]
Lake Hollywood Drive is pretty narrow in places, with either limited or no parking rules along several sections. Check signage before you park, including any signs you may pass with an arrow (indicating parking rules for extended segments of road).
There may be parking near the end of Innsdale Drive, parts of Tahoe Drive, and the part of Canyon Lake Drive adjacent to Lake Hollywood Park, although I haven't hiked around Lake Hollywood Reservoir in a few years, so I'm not sure if accessibility has changed. I do know that the exit off the 101 for Barnham was removed, which makes the slightly more complicated route to this area necessary. They have also changed street parking rules a few times in this general area, because it can get congested.
It should go without saying, but if you do try to hike from the area south of the Hollywood Sign, don't block any driveways and follow the rules. The bigger pain in the butt you make things up there, the more likely additional restrictions and stricter enforcement will follow.