The main thing to ensure is that she knows
why you want to get her alone. This will take a bit of balls, just inviting her for gaming is a lower pressure approach but is likely to be interpreted as just a friendly invitation, even if she can connect the dots herself she could easily find it creepy that you present it as one thing when it's another.
How I would approach it is this. When she comes over for regular gaming greet her with a genuine compliment -- you like her eyelashes, her nailpolish, her outfit, her style, something like that. This lets her know you notice her, as more than just a gamer. If you can get a bit of touch in while you greet her (touch her on the arm while ushering her into the house or similar) then so much the better. But you have to use calibration because some of these things might be totally inappropriate, for instance if she arrives in a group, the keyword is being discreet here (I say this from experience having burned myself with at least one workplace girl by putting the moves on her in the presence of friends, etc, when she was clearly attracted originally).
Now you have to find opportunities for conversation, if she arrives early then great, or maybe you can ask her to help you with the refreshments at the appropriate time and take her into the kitchen, or perhaps even during the game you could set up a bit of flirting "okay we are gonna join forces and kick these guys' arses" or similar... but basically you want to ask her as much as possible about herself, how was her day etc, what has she been doing lately and then use this information to get her talking. You also need to mention sex or hot guys or dating or
something, as often as possible, just working it into the conversation, so she sees you as a sexual man. All of this could take a few sessions, just take your opportunities where you can and use them (I'm currently running a similar game on a girl who works at the zoo, I only see her on Wednesdays at the ticket booth but things are progressing quite nicely, it's just frustrating when co-workers are present).
As to asking her out, you could try a yes-ladder, wait until she's been telling you some important and emotional stuff about herself, her hopes and dreams etc, then say "this is a really interesting conversation, do you agree?" "yes" "I find you intriguing and I would like to know more about you, do you feel the same way?" "yes" "in that case why don't we exchange numbers and go on a date, or you can just come here to hang at a more convenient time if you're cool with that"... hmm I'm just making this up but I think it would work, provided nothing goes awry in your yes-ladder. Of course I may be preaching to someone who already has good game and your style might be totally different so don't take it literally, just throwing ideas out for you
cheers, Ray