I believe it is a state of mind before anything else. And a state of mind can be acquired, or refined.
Yes, you can become Alpha, but it starts from the inside out, not the opposite. When I go out I can litterally point out guys that act like they're the shit but are built like card castles inside their heads. You could walk up to them and plant a pebble in their head and it would instigate doubt for weeks.
I don't give a shit if I'm alpha or not, but I take decisions when they are needed and I walk with my head up. Sometimes I decide, sometimes I follow. I don't analyze it, it's a waste of time.
Another thing to consider is that someone that projects leadership and power over the others in a group might be seen as the alpha of that group for instance, but the discreet guy that doesn't want to be told what to do and decides to break off because he knows he is better off on his own, and walks away completely detached of what others think of him is equally alpha, but it's not on the same level.