How to build a career acting in English, if English is not your native language
I am Romanian by birth and the aspect of speaking English as a native has always preoccupied me as it was a way out of being typecast as an East-European.
From my own experience as I lived in the UK for 7 years now, there is no other way to master a language than by studying continuously, training your ear with RP private lessons (if you can afford them), listening to BBC, to tapes with RP and songs sang by British singers (it certainly helps to have variety in the mediums you use to learn). The other extremely important aspect is to mingle with British people in your everyday life.
Another important thing I found in learning a foreign language as your own is to become acquainted with Phonetic transcription as it gives you a direct answer to how to pronounce a word when you have doubts and wish to know the RP for it.
There are certain expressions you might want to write down and remember which go with the typically English ways of expressing certain things as well as English mannerism that also create the identity which people will identify as being English which you might need to develop an eye for.
And if you are really bold (which in my opinion you need to be as an actor - it also helps to free yourself of all inhibitions - get yourself on a Tutoring site and teach English as a Foreign Language. That will help you become aware of the mistakes people of other nationalities (including your own) typically do, which in turn will help you whenever you might be asked to speak English with an accent, once you reach the point you sound like a native English.
All in all, being receptive, listening to people around you, making notes of things you might not understand or you might find surprising (linguistically or culturally)are equally important and will help you in mastering the English language to a degree that people won't take you for a non-British/American (depending on what accent you want to go for). Oh, and one more thing, crucial in fact: be extremely patient and kind to yourself. There is a lot of time and exercise involved in obtaining the desired result and a lot of trial and error too! Keep working on it. Never give up the hard work and I promise you you will get there.