Testing Google Play
Login to Google Play publishing/developer console.
The .apk needs to be (published) as an alpha app (in the alpha tab) for testing, not uploaded as a draft. The app will need to be ready for publishing, which means that all info in Store Listing and Pricing & Distribution need to be filled in before testing can take place.
In the top right corner of the app's listing in the Dev Portal, there's a drop down menu to 'Publish'. If it isn't ready to publish there'll be a dropdown option for Why Can't I Publish? which will provide info on what settings need to be filled in.
The way of 'approving' accounts as testers has changed - see http://developer.android.com/distribute/googleplay/developer-console.html#alpha-beta. Create a Google+ Community or Google Group - and add the email created for this group under 'Manage list of testers' for Alpha Testers in the APK section of the menu. Once done, under 'Share the following link with your testers.' In the same popup a link will appear - this is the link to share with testers.
When they follow this link they can opt-in to testing from it and will be given a download link to download the .apk on their device/s (on which you must be signed in with the same primary google account).
You must also create license test accounts for authorised users (so that they are not charged for test purchases). In the Developer Console, go to Settings > Account details, then in the License Testing section, add the addresses to Gmail accounts with testing access field.
Single issue purchases count as 'test' purchases and will not be charged. Subscription purchases need to be cancelled via Google Wallet under 'Bills and accounts'. You can also cancel via: https://play.google.com/store/account for the user.
Note: we have discovered one issue with this, which is that when an updated version of the .apk is uploaded to Alpha, the download link doesn't update immediately. Thus someone might have a link that seems to be to a newer version of the .apk but isn't. Therefore, advise filling in fields such as 'What's new in this version?' to help identify versions correctly.
Note 2: If opening links in Chrome to download, ensure you're not logged in to a different Google account in Chrome