I thought about starting to give her the smiles in hallways when our paths cross and stuff so when our next conversations occurs I can more easily ask her out. Good?

I don't think it's nearly enough, but it's better than nothing.

So I should just do something that's the pure opposite of my character? Isn't that just faking it? It's basically fake it 'till you make it.

I don't know what to tell you. Society values extroverts and people who are charismatic or good story tellers. As comfortable as you are spending time alone not trying to impress anyone, I'm sure if you could just flip a switch and be a social butterfly whenever you wanted that you would. Well there's no switch, but you can certainly become that person if you want, but it's a matter of effort. You don't big and muscular without going to the gym and you don't learn to be social without socializing.

Have you ever considered that extroverts aren't just naturally good at socializing? It's more likely that they seek out opportunities to spend time around other people and have therefore accumulated a life's worth of experience in social situations. The average person doesn't have any innate advantage over you, just experience. If you don't want to put in the time or effort then that's your call, but it will make dating much harder for you. The sooner you start, the less behind you'll be.