Tell her you play games. If you hide it, she'll wonder why and that will lead to problems. (Telling her 'because I know you don't like them' will probably make her think you don't trust her, another 'deal breaker')
Don't try to force games on her, best case she'll just say she's not interested at all, worst case, you give an impression that you won't accept her unless she plays games, she ends up hating you and all games because of it, and you break up in a bad way.
Since you also said she's neutral on the issue, don't worry too much. For you to know this, you'd have to have bought the subject up before. If you continue to play games, and not shaft her whenever you are playing games, you will get to keep a great passtime, and will likely earn her respect. (Steriotypes work in your favour here, remember all the 'If a guy talks to you whilst playing COD he really likes you/is just dead facebook likes? by proving this isn't true, you show yourself as above the rest [hopefully])
If she's not OK with you playing video games, you have two options. Either way though, you must take a firm stance. Don't act all depressed or anything, no sad acts. Just be firm and either:
a) Tell her that you will give up video gaming and actually do (If you decide to keep playing games after telling her you'll stop, your bound for trouble). This will hopefully earn her approval and respect, and show her how much she means to you.
b) Tell her that you will not give up gaming, it is a part of who you are, and if she can't accept that then she's not right for you. Chances are, its true. If she can't accept you gaming, she probably isn't the right girl for you. This is also, by the way, effectively you 'breaking up' with her, rather than the other way round, so hopefully you won't hurt as much if you choose this. Best case, she sees that what you said is true, and decides that maybe she wasn't trusting you enough and you stay together, and you get to keep gaming. Worst case, you have a mature 'break up' that you can both hopefully accept.
I will make a couple of things clear under no circumstances should you:
- Hide the fact that you game from her
- Lie to her
- Try to coerce or force her into gaming
- Not make a decision on which of the above two 'She hates gaming' solutions posted above
- Let gaming dominate your time
-Flood her with games, try maybe one or two a week at first. If she likes them, gradually build up.
If you want to get her into gaming, pick a game that she might enjoy* and play that casually whilst around her (Only when talking or watching TV [Time wasting activities], never when doing anything else). If she looks interested, ask her if she would like to have a go/learn to play, if she looks bored, stop and give her some attention. Note: even when playing this game, look at her when she is talking and you are talking to her, show her she takes priority. Also, I would find asking her if she would like to play a PC game with you when she hasn't shown interest in it is inadvisable, not a 'DO NOT DO', but inadvisable.
*If she likes puzzles, Portal or Prof. Layton or any similar game [even Legend of Zelda has some elements of puzzle, but it wouldn't be my first choice], if she doesn't like puzzles, but enjoys 'deep thinking' per se, try turn based strategy games [Civilization 4 probably the best I can think of {Not 5 in my opinion}] if she likes deep narratives, try some Bioware games [Or any other company with equally as compelling stories, Bioware is the only one I know of ATM, open to any suggestions in PMs] [Mass Effect 1 and DA:O are some fairly recent ones, and IMHO have better stories than their sequals] ect.
Hope I helped.
Edit: In the time it took me to write this, around 10 people posted most, if not all, of my points individually. Also, if she plays a major part in your life, and she's a new girlfriend of 1 day or so, maybe think about how much a 'Major Part' is, and if she should take that up when you don't truly know her.
For laughs, I'm going to go with a number of people on this thread and say the Lego games, my favourite being Lego Starwars 2. Absolutely hilarious, and it got my dad back into gaming (Until we finished it)