A way of encouraging the user to submit their email, is through "good" association. By asking users to share their email after they've done something fun / entertaining / beneficial, they're much more likely to share it, because their brain will associate that action with the feeling that they just had or want to have (assuming they've clicked on a video and you ask right before or after).
Say they've finished reading an article and they return to the home screen. You could display a pop-up with something along the lines of, "Enjoy reading these articles? Enter your email address to keep you updated on major features!"
If you have a rating system you could always detect when users rate something really good and ask them then.
Another way to do this is, to also display it after a video with something like, "Enjoyed the video? Help us improve our collection! Enter your email address in case we need suggestions or feedback!"
You could also provide unlockable content, in exchange for their email address. (I.E. You could display only the top 10 least news articles, but if they share their email address the app will display the latest 50, etc).
To answer your question about the skip button, I think that should also be fine. I don't see why Apple wouldn't allow that, since you're not forcing it on users.
NOTE: I strongly suggest that you're transparent as possible and let them know why you're collecting their email addresses. If you're not doing it for a bad reason I think most people wouldn't mind, especially if you make them feel good about doing it. So just be honest and use these tips to help increase the number of users willing to share!
For more information on the theory of association here's a good place to start: http://en.m.wikipedia.org/wiki/Association_(psychology)