You can post a free trial version of your app in Google Play and when the trial expires then they will be required to download a license file from your website.
This file will only remove the trail and let user use the app. You can add a notification when after 7 or so days of trial to purchase license. when they press okay then they would be directed to your website and download the license apk file after they pay.
And then after you post an updated version of your app in Google play. They will be able to update too and they will not loose their full version of the app.
Please make sure that you describe your app correctly in google play, and state that your app is free only for some days. and do not post a link to a full version app in your description if you are planning to host it on your website.
You can also integrate in-app purchase with the help of Paypal. This documentation teaches you how to do it.