There are 4 options really.
Build it yourself. Offer conent to email subscribers, such as music from your game, artwork, desktop and mobile backgrounds, exclusive content, a dev or artwork blog. People like emails with content they are interested in. The problem here is everyone on the list already knows about you. It only becomes a good way of marketing when you have multiple things to sell.
Build it with a partner. Find other devs in a similar position and make it clear when they sign up they are signing up to a shared newsletter. We have a client who shares a newsletter with 3 other mobile developers and their list has grown massively since partnering up. The biggest advantage is this is a way to market to people who do not know about you already.
Get a marketing agency to send emails for you. A Marketing agency that specialises in video games, mobile apps, gambling, or any related nice will normally have ways of acquiring decent email lists, either through partners and other clients in the niche, or ones they have built themselves. They can send emails to a a couple hundred thousand to a couple million people interested in gaming.
Purchase an email list. I do not recommend this to anyone who doesn't know what they are doing. If you buy the wrong email list, you could be using stolen data, or spamming without even knowing. It is possible to buy legit lists where you partner as a third party, and get the list of people who forgot to click that they didn't want third parties from having their data. You can then send a massive amount of emails yourself. If you get a list of 5,000-10,000 plus you will probably need an email service other than mailchimp, they are quite vigilant about large lists. Something like Silverpop, phpList or Bost work, there are plenty of others out there but those are the only ones I have used. For smaller lists, MailChimp is definitely the best unless you want a self hosted service like Bost of phpList.
Source: I work for a marketing agency that works primarily with gambling, but recently have been branching out to mobile games.