Also, you should understand (especially in the UK) that people don't all speak in the same way. Age, gender, social class, geographical region and many other factors all play a part in determining the way someone speaks. You must gradually find a way which is unique to you and which allows you to express the identity you wish to express.
But let me add that what I think largely characterises a native speaker's speech is the rhythm with which she speaks. If you can imitate the stress-timed rhythm of native English-speakers, you will sound a bit like one. Let me also add that it is extremely rare for a non-native speaker to ever achieve a fully native accent, so it's not a useful objective to have, if that is your goal.