"No hugs! I'm British. We only show affection to dogs and horses."
- From 'What A Girl Wants', I think :P
In my opinion, to be classified British you should have been born here/lived here for X number of years/hold a British passport.
But! To truly feel British I think you have to like the culture and food too, whether it be Fish & Chips, ceilidhs, moaning about the weather, increase in prices, tea, scones, etc
You don't have to be born here to be British strictly. Example: My dad was born in Kenya to a British family, moved to South Africa when he was 2 and only moved here when he was 24. He grew up in SA and considers himself South African. He did everything that a teenager born in South Africa would do, therefore he is South African.
However (this is an invented example), say a family from Russia move to the UK. They have a kid who grows up in the UK, however the family is strict and the only culture which any of them grasp is the Russian culture, then they would not be British, because they aren't like the people here. Make sense? They wouldn't have been intergrated (I'm not saying you have to be 100% intergrated, but you get what I mean).
Of course you can say you're British if you're born here, but you're only really British if you're ancestory has been British for a substantial period of time or if you are British through your lifestyle etc.