Legally changing your name is a pretty big step, most people don't do that until they are ready to come out and live full time in their proper gender role. There's nothing that says you can't start using "Alex" as your nickname though, and get everyone used to referring to you by that name, including yourself. It varies state to state or if you live outside the US, but where I am there was no proof required at all. For the "reason" on the name change form I simply put "common usage" since everyone already called me Chloe anyways.
Alex is one of those names that could be either gender- I knew a girl in high school whose name was Alexandra but she went by Alex to everyone, including her teachers. If you like the name there's no reason you can't start using it now, unofficially. Later if you decide it's what you want you can get it legally changed then.