How do you become friends with someone famous?

To become friends with someone you need to be able to meet them and talk to them socially. The problem with people who are famous is that they like (need) to maintain their privacy and therefore restrict who has access to them on a social basis. The most neutral approach would be to write to them and say you would like to meet them.
However, the very nature of your question 'someone famous' indicates that you are not actually interested in the famous person on a personal basis (they could be anyone as long as they were famous) but simply for your own aggrandisement. You would be the sort of person a reasonably minded famous person would want to avoid socially.