Los Feliz is considered an excellent LA neighborhood to live in. Like in most of Los Angeles, there are very exclusive and expensive areas, for example north of Los Feliz Blvd, then there are very nice area but more reasonable in price, near Franklin Ave, and some more artsy (less expensive) areas. The mix makes Los Feliz a very fun area to live in! People I know that live there absolutely love it!
As other realtors mentioned, what will be Safe to you will be relative to your own standard. You can get statistics on line or with the local police department.
Statistic will be different depending on the more specific neighborhood or even streets that you may be interested in. With a little bit more details about your specific search regarding size, price point, and your priorities, I'd be able guide you to the best area for you.