1. Write a great screenplay in one of the genres they favor (see the previous post). This spec script (meaning you're writing it on speculation--that is, for no money upfront) will be what opens doors for you. It has to be the best you can do.
2. The first set of door it must open is the one that leads to getting a US agent to represent you. Some UK agents do have arrangements with US agencies so it's possible to get in that way, but it adds another layer.
Most of the bigger agencies say they don't look at unsolicited scripts. That means you have to find a way around that, either by getting their attention with a great query letter in which you offer to sign a release form, or getting a recommendation via a writer they already represent, or by some clever strategum you figure out yourself.
(For help with inspiration, see my book "Do Something Different," published by Virgin Books--it's not about writing but it contains 100 case studies of how people used clever, inexpensive ways to market themselves and their products, and you could adapt some of those to this situation.)
3. Tell the agent you frequently go back and forth between wherever you are and Los Angeles (that's where the agent should be, ideally). This will make him or her less hesitant about setting up meetings with you and any of the producers who show interest in your script. Be ready to finance that first trip yourself.
4. Don't assume that once you have an agent everything will be done for you. You still need to educate yourself about who is buying what, who is who at the studios and production companies, etc. Variety and the Hollywood Reporter, although both suffering in the current shift away from print advertising, are still your best bets for US information. Keep the agent informed of any initiatives you plan to undertake yourself--it has to be a partnership. The agent doesn't want to be embarrassed by approaching someone on your behalf only to find you've already been there and got a rejection.
5. Finally, you need a load of good luck and good timing.
Hey, I didn't say it was going to be easy.