If your installing from your own site, don't increment the version code and name of new apk your uploading, use the same code and name of already existing apk, it will replace the existing one so no need to uninstall the app.
Try by enabling the "install app from unknown source" in device settings
Best way to do Even if your using the app for internal purpose you can sign the app using live keystore(not debug keystore) and upload it in your own site.
Another way to avoid getting your scenario:
If you want only some particular users to download the app that is the reason if you have hosted your app in your own site means, google provides an option to do this for testers where you can add their email in alpha testing or beta testing so that whenever you upload an apk to playstore those users who you have added will be notified and can update the app without uninstalling the existing(this wont be visible to all users who you have not added).