How to update webview without using the Google Play store?
The Google WebView application is very different from regular ones. You cannot use Package Installer to install or update it in a common way. You must do it yourself (As for my Lollipop 5.1.1).
Here's the way I've done it several times. Run in a root shell. You may need busybox for unzip (or do it with a file manager).


#!/system/bin/sh
rm -r /data/app/com.google.android.webview*
mkdir /data/app/com.google.android.webview-1
cd /data/app/com.google.android.webview-1
cp /path/to/your/webview.apk ./base.apk
unzip base.apk 'lib/armeabi-v7a'
mv lib/armeabi-v7a lib/arm
# Do the following 2 lines only if you're on an ARM64 ROM
unzip base.apk 'lib/arm64-v8a'
mv lib/arm64-v8a lib/arm
#
find . -type f -exec chmod 644 {} +
find . -type d -exec chmod 755 {} +
find . -exec chown 1000.1000 {} +


Alternatively, you can try creating directory /system/priv-app/WebView and put the new APK there. Remember to get rid of the old one.