>>1 Please consider what I have to say. I don't think the advise so far has been very useful.
First off: You seem to actually know how to play Magic, and thus it would be incredibly hard to fake it, so rather than ask him to teach you from scratch, emphasize that what you're wanting help with is strategy. This won't seem too out of place, and should be far easier to fake, if you don't actually need the help.
Second: Please do avoid that overly touchy, flirty, etc stuff. It can actually put a guy off because you seem, well, pathetic.
Third: The best thing to do is to ask non-personal questions and honestly just get to know him while playing. Treat it like dinner. Keep the game going, but at an easy pace. When you find things that the two of you appear similar on, smile, show interest in the subject, and point out that you're glad/think it's cool you agree.
When you disagree, feel free to point it out, and why, but don't push the subject into an argument if it appears to be going anywhere beyond light-hearted.
With this casual communication, the natural signs of your attraction such as eye movement, interest in him, body posture, etc should show up. Toward the end of the game (Though I don't know that I'd say to wait until it's over, that would also be ok), ask if you can meet him somewhere or if he'd like to go out on a date with you, if you think he is clearly interested and hasn't asked himself. Alternatively, if you think he's interested but not so shy that he'd be hesitant to ask you out, ask him if he finds you attract, and smile when doing it. It should unfold easily from there if he does like you, and, if not, you can simply keep smiling, nod, and then go back tot he game.
If you've got guts, you can ask him why, and consider his thoughts honestly to help you improve in the future. You could even ask if you've made him feel uneasy 'tonight'. Take it as a learning experience and also know that sometimes people just don't fit together.
Finally: If it doesn't work out with him, get in touch and perhaps we can hang out. :D