It depends on the person. Some will use it as basically a replacement for their actual name, others will use for only for business with people who can't speak Chinese.
Interestingly, there doesn't seem to be as strong a correlation as you'd expect between a person's use of an English name and their command of the English language.
I used to work for a startup in Shanghai where no one spoke a word of English, except for the English names of the other employees. Office chatter was entirely Chinese, with a sprinkling of "Tony", "Leon", and "Francis." At the other extreme, I had a friend who was from Chongqing, but grew up and went to school in the west, who refused to take an English name at all because she didn't see the value in going by a fake name to all her friends.*
However, by her own description, this was somewhat unusual, as it meant that people called her by her full given name all the time. And according to her (this could be wrong, I'm just repeating it, and its hard for me to verify as I'm not Chinese), Chinese people generally address each other by some kind of nickname. So bringing it full circle, an English name is not only kind of cool, but doubles as a nickname, which was already a part of the culture anyway.
*I can see the merit in this. If you're going to befriend someone, especially someone of a different cultural background, telling him/her your name is basically the first chance you get to introduce them to your culture and language. But I digress.