How to get un-solicited scripts seen by a literary agency
I have done this and got my screenplay read by two literary agencies.
The best thing to do is to get a copy of the Writers and Artists Yearbook, which you can get in any library in the reference section or on Amazon for £13.39. In it there are loads of articles and advice on writing etc, but the ones you want are 'Letter to an unsolicited author' and 'How to attract the attention of a literary agent,' and these will tell you the best ways to go about it. Research who to write to carefully, as not all agencies will want screenplays, and some may want only specific genres. This is real leg work, and may take days, but it should be done, and a full list of agencies will be in the book too.
Now most agencies will state 'no unsolicited scripts please'. You have to take this to mean 'write to us first.' This is pretty much following the norms of writing to an actors agency. So when you've chosen who to write to, in your letter tell them the basic plot, why it stands out from the crowd (fast paced dialogue for example), who the script might be aimed at (this will help with selling it, which they are hopefully going to be doing), and some information about your writing history (I had none!). It's very important to tailor your letter to who you are writing to as this shows you've done your research and know what you are talking about, as in if it has Mike Leigh elements to it, reference that in your letter if that agency you are writing to represent Mike Leigh. Try to keep it concise as well, but that should be no problem for you are you're a writer aren't you? And make sure you follow the rules about sending SAE's if you actually want a reply but also included all contact details (got a letter and an email reply), and once sent, be prepared to wait a while, but I'd say you could chase your letter up after about 2 months if you haven't received a reply yet.
Joy of joys, they've replied and asked for a copy of the script! Oh crap, it's not finished yet! So if you do get that letter begging you to see your work, make sure it's ready and waiting to fly out of your door - the last thing you want is to then have to take a week to rush finishing the thing and short changing yourself! What you WILL want to do though is go over it and make another final draft, so feel free to do that as it will only take a few hours if you are near enough satisfied.
Then once sent, don't be disappointed if they are not interested. It is a very, very long game, so either use other contacts, like BBC Writer's Room, and tell your acting agent as well if you have one, and they might know people who want a look at it. In the meantime, get onto writing the next thing, and once that is finished and as perfect as can be, send a letter about that one as well. And keep it up; if you keep producing work they will be impressed, and it's only a matter of time before they cave in with your soon to be massive portfolio of work!