Whom to send your cover letter/resume (and how to get their contact info) is probably the biggest hurdle. Different productions will designate different people to supervise interns. If you can find a friend of a friend of a friend, you'll be in the best shape. If not, I would do as much research as you can to figure out who is in charge of the show on the production company level, studio level and network level. If you can find the production office number, you might be able to simply call and ask who handles interns and if you can submit a resume for an internship, and if you sound like you're smart and not psycho, it just might work. (Internships are much easier than jobs in this way.) You might also be able to find some info from The Hollywood Creative directory, IMDBPro, or (if you have a friend with access) the expensive StudioSystem. Also, one of the upsides of attending a college's LA program is that the school will already have some contacts. Keep in mind that many internships will require you to receive college credit -but if money is no option, you might be able to skirt that as a graduate by enrolling in a local college class. (Yes, you'll literally be paying to intern - but that's how it works for undergrads too.)
I find that for Hollywood jobs, the shorter your cover letter, the better. I usually don't even write a separate cover letter document unless the posting specifically requests otherwise, because then you have to write an awkward email in addition to your cover letter. I combine them into a cover email, and I usually just write a quick note about how I found the job posting (if that's relevant - for example, if the person you're emailing knows the friend who told you about the job, that's important), why I want to work there, and why I'd be a good fit (experience, etc.). If you're applying from a posting, tailor what you say to that posting. Don't be too stuffy or effusive - but poor grammar and emoticons are also a bad idea. Generally, the stronger your connection, the less you need to write to explain your application. (Sometimes I literally just forward my resume.) But in your case, I think just a few sentences is fine. Remember that these people are busy and may have to sort through hundreds of emails and resumes. Here is one email I wrote that resulted in me getting an interview:
I saw the posting for an assistant position and I'd like to apply. Working for a lit agent for over a year has provided me with a breadth of knowledge about Hollywood, and also helped me hone my skills as an assistant. I feel I'm ready to make the move to working for a writer as that is my ultimate goal. I am ambitious, organized and computer-savvy, and I thrive on juggling many tasks at once. I also have a lot of experience with script coverage and would
be happy to send samples.
I have attached my resume. I can be reached at (phone) and (email). I look forward to hearing from you!
Here's a much shorter one that I used when I had a really strong personal connection. This also resulted in an interview.
(Specific person) forwarded me the posting for the 1-hour cable legal dramedy showrunner asst position and I'd like to apply. I've attached my resume.
Thanks a lot!
Unless you think you are qualified for something more, I would ask for an internship position. Lower level jobs on a show include on-set PAs and office PAs - but I bet almost all of those people have already been interns somehwere (or have really strong personal connections).