How to get work in the US as a UK trained actor

This can be a time consuming and lengthy process.
A good way to start is to look at some of the casting director/agency workshops that some companies hold in the US. These are not just designed for British actors but alot of US actors attend these classes too.
They are not like the ones in the UK, these casting directors hold these workshops to find the next character in a series they are currently casting, they love fresh new faces and dedication. Make sure you do your research of who is casting what and when.
Have a look at TVI Actors Studio New York or LA. They also offer courses which last around 1 week and help you with paper work and and other bits of information you may need to know, including getting an agent and a Visa.
In the US you cannot work in the industry unless you are with the actors union, it is like our EQUITY but over in the US its a must. Be careful, if you work without being in the union and the government find out, you will not be able to work in the US again. Its just not worth it.

Another way and a way which most successful Brits continue to work in the US, is to send a video casting for a project you are suitable for, from there the company will sponsor you and all the paper work will be sorted through the company hiring you. I've started noticing alot of self tape castings on spotlight recently for British actors to work on American series, make sure you tell you agent to submit you.

If you have never been to LA or New York before, I would take a two week - month holiday, meet people, connect, see what's what and who's who.
Its all about networking in the US.
Also have a look at 'Brits in LA' facebook page. I've seen alot of very interesting and informative posts.