Information on hiking to the mountain top (Dante's Peak) overlooking Griffith Observatory can be found at these links:
http://www.yelp.com/biz/griffith-observatory-hike-los-angeleshttp://www.webworkit.com/latrails/griffithdante.htmhttp://baldrunner.com/2009/06/11/dantes-peak-griffith-park/There are various trail-ways leading up to Dante's Peak. But the easiest way up is to start at the trail head directly in front of the Observatory, right next to the parking lot. You don't really need to hike all the way to the top to take good photos of the Observatory. Actually just 1/2 mile up the trail is probably good enough. But the further you go up, the more of LA cityscape you can get into the frame. On a clear day, you could see Catalina Island from Dante's Peak.
If you are going to stay at the top taking cityscape photos after dark, make sure to bring a good flash light with you and be careful hiking down the mountain in darkness.
Have fun !!