You can work for them - find people, ask for work, do missions, repeat - but you can't actually join (unless your asking about Fallout 2, where you can become a ranger).
It seems wierd, but you cannot join the NCR. You can only work for them until you reach the Second Battle for Hoover Dam, then you decide if you want an independant New Vegas or a New Vegas annexed and controlled by the NCR. Remember, your actions control the fate of New Vegas years to come. You just can't join the NCR, and I know, it sucks.
If you help the NCR annex New Vegas and continue to live there you'd theoretically become an NCR citizen along with everyone else. By then the game is over, of course.